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1. Find parametric equations describing the line segment from P(0,0) to Q(7,17).
x=7t and y=17t, where 0 ≤ t ≤ 1
x=7t and y=17t, where 0 ≤ t ≤ 1
2. Find parametric equations describing the line segment from
![{\displaystyle P(x_{1},y_{1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c40f1c73dac610d345c535db1cc111415a14d97c)
to
![{\displaystyle Q(x_{2},y_{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff3be1197a61155a6be18e775a94dd250bd4cf9)
.
![{\displaystyle x=x_{1}+(x_{2}-x_{1})t{\mbox{ and }}y=y_{1}+(y_{2}-y_{1})t,{\mbox{ where }}0\leq t\leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/566149b381b475c92d014bb9ef9e74af723a1164)
![{\displaystyle x=x_{1}+(x_{2}-x_{1})t{\mbox{ and }}y=y_{1}+(y_{2}-y_{1})t,{\mbox{ where }}0\leq t\leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/566149b381b475c92d014bb9ef9e74af723a1164)
3. Find parametric equations describing the ellipse centered at the origin with major axis of length 6 along the x-axis and the minor axis of length 3 along the y-axis, generated clockwise.
![{\displaystyle x=3\cos(-t),\ y=1.5\sin(-t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29d7a525a585a07d2f091f2f8d0b90e6dadcbe24)
![{\displaystyle x=3\cos(-t),\ y=1.5\sin(-t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29d7a525a585a07d2f091f2f8d0b90e6dadcbe24)
21. Find an equation of the line y=mx+b in polar coordinates.
![{\displaystyle r={\frac {b}{\sin(\theta )-m\cos(\theta )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d8013ec941e02a4d9857111a5815d43308eae6)
![{\displaystyle r={\frac {b}{\sin(\theta )-m\cos(\theta )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d8013ec941e02a4d9857111a5815d43308eae6)
Sketch the following polar curves without using a computer.
22.
![{\displaystyle r=2-2\sin(\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89eb7d1cea850b34322e26aeb873f19dd1cf5dd6)
23.
![{\displaystyle r^{2}=4\cos(\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd40021e28f81795d0ed3d91ded3c59eb9b8c95)
24.
![{\displaystyle r=2\sin(5\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47703934f0d67346a46462299eb7e32cea6b4c06)
Sketch the following sets of points.
25.
![{\displaystyle \{(r,\theta ):\theta =2\pi /3\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1836249e0f007444d179f372e524c17be93855b2)
26.
![{\displaystyle \{(r,\theta ):|\theta |\leq \pi /3{\mbox{ and }}|r|<3\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2249db3f3e74fd825ddd90e811e091f5d8748c4b)
Find points where the following curves have vertical or horizontal tangents.
40.
![{\displaystyle r=4\cos(\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62ee4e5618c074debbd695aa5915ad54d2f8144d)
Horizontal tangents at (2,2) and (2,-2); vertical tangents at (0,0) and (4,0)
Horizontal tangents at (2,2) and (2,-2); vertical tangents at (0,0) and (4,0)
41.
![{\displaystyle r=2+2\sin(\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b267ee749c4bf29cf381fa49b60e942eaba24807)
Horizontal tangents at (r,θ) = (4,π/2), (1,7π/6) and (1,-π/6); vertical tangents at (r,θ) = (3,π/6), (3,5π/6), and (0,3π/4)
Horizontal tangents at (r,θ) = (4,π/2), (1,7π/6) and (1,-π/6); vertical tangents at (r,θ) = (3,π/6), (3,5π/6), and (0,3π/4)
Sketch the region and find its area.
60. Find an equation of the sphere with center (1,2,0) passing through the point (3,4,5)
![{\displaystyle (x-1)^{2}+(y-2)^{2}+z^{2}=33}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29b2561d00f6211ec4e2b33d3026edf073163fdb)
![{\displaystyle (x-1)^{2}+(y-2)^{2}+z^{2}=33}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29b2561d00f6211ec4e2b33d3026edf073163fdb)
61. Sketch the plane passing through the points (2,0,0), (0,3,0), and (0,0,4)
63. Find all unit vectors parallel to
![{\displaystyle \langle 1,2,3\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7035fe004039b3f11a88be5758a290c80f74315)
![{\displaystyle \pm {\frac {1}{\sqrt {14}}}\langle 1,2,3\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/21a0ce14c1d8bd8220dfdd1e4fbe9615e89a3ee1)
![{\displaystyle \pm {\frac {1}{\sqrt {14}}}\langle 1,2,3\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/21a0ce14c1d8bd8220dfdd1e4fbe9615e89a3ee1)
64. Prove one of the distributive properties for vectors in
![{\displaystyle \mathbb {R} ^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5)
:
![{\displaystyle c(\mathbf {u} +\mathbf {v} )=c\mathbf {u} +c\mathbf {v} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/864a42744bed5d422284ec8c5b6b441afd443543)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle \begin{eqnarray} c(\mathbf u + \mathbf v) &=& c(\langle u_1, u_2, u_3\rangle + \langle v_1, v_2, v_3\rangle)\\ &=& c\langle u_1+v_1, u_2+v_2, u_3+v_3\rangle\\ &=& \langle c(u_1+v_1), c(u_2+v_2), c(u_3+v_3)\rangle\\ &=& \langle cu_1+cv_1, cu_2+cv_2, cu_3+cv_3\rangle\\ &=& \langle cu_1, cu_2, cu_3\rangle + \langle cv_1, cv_2, cv_3\rangle\\ &=& c\mathbf u + c\mathbf v. \end{eqnarray}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle \begin{eqnarray} c(\mathbf u + \mathbf v) &=& c(\langle u_1, u_2, u_3\rangle + \langle v_1, v_2, v_3\rangle)\\ &=& c\langle u_1+v_1, u_2+v_2, u_3+v_3\rangle\\ &=& \langle c(u_1+v_1), c(u_2+v_2), c(u_3+v_3)\rangle\\ &=& \langle cu_1+cv_1, cu_2+cv_2, cu_3+cv_3\rangle\\ &=& \langle cu_1, cu_2, cu_3\rangle + \langle cv_1, cv_2, cv_3\rangle\\ &=& c\mathbf u + c\mathbf v. \end{eqnarray}}
66. Find all unit vectors orthogonal to
![{\displaystyle 3\mathbf {i} +4\mathbf {j} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7980e9e0edd2c5642a4ca348fc246c1a368b6aec)
in
![{\displaystyle \mathbb {R} ^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5)
![{\displaystyle \left\langle {\frac {4}{5}}c,-{\frac {3}{5}}c,{\sqrt {1-c^{2}}}\right\rangle ,\ c\in [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec58739fb84b755913f1b1240cecc19cacfcaa51)
![{\displaystyle \left\langle {\frac {4}{5}}c,-{\frac {3}{5}}c,{\sqrt {1-c^{2}}}\right\rangle ,\ c\in [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec58739fb84b755913f1b1240cecc19cacfcaa51)
Find
and
80.
![{\displaystyle \mathbf {u} =\langle -4,1,1\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3cc81e26260bf95572ed05c035c466d72d7fca5)
and
![{\displaystyle \mathbf {v} =\langle 0,1,-1\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/446af0e93cd08418eab3e6d072ae86473aadcd16)
![{\displaystyle \mathbf {u} \times \mathbf {v} =\langle -2,-4,-4\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/343ae0cb2fbb920a0bb6d60c47dceb28d233ed56)
![{\displaystyle \mathbf {u} \times \mathbf {v} =\langle -2,-4,-4\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/343ae0cb2fbb920a0bb6d60c47dceb28d233ed56)
81.
![{\displaystyle \mathbf {u} =\langle 1,2,-1\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ea14a685318f6b12f349ce3128623b1199278a)
and
![{\displaystyle \mathbf {v} =\langle 3,-4,6\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f28ec1e044d6b3b86b42f67e33a062be68e8cee)
![{\displaystyle \mathbf {u} =\langle 8,-9,-10\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd2cc11ee50238500d1fe88674431af61fd626a6)
![{\displaystyle \mathbf {u} =\langle 8,-9,-10\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd2cc11ee50238500d1fe88674431af61fd626a6)
Find the area of the parallelogram with sides
and
.
84. Find all vectors that satisfy the equation
![{\displaystyle \langle 1,1,1\rangle \times \mathbf {u} =\langle 0,1,1\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a5ab6d6ab73a6ad45be8cbcd11b27c164a49186)
85. Find the volume of the parallelepiped with edges given by position vectors
![{\displaystyle \langle 5,0,0\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/890ffd433a0f58e918c9acffb407ce90010bf91e)
,
![{\displaystyle \langle 1,4,0\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7340622045dacc04f3ea01ee2afd99c50730fadb)
, and
![{\displaystyle \langle 2,2,7\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcbb9a7383295fdb038fa6a4d859a2ffca42dbed)
![{\displaystyle 140}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75434a8ecb5f74f1ac5817e8d7df736dda06fbb8)
![{\displaystyle 140}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75434a8ecb5f74f1ac5817e8d7df736dda06fbb8)
86. A wrench has a pivot at the origin and extends along the
x-axis. Find the magnitude and the direction of the torque at the pivot when the force
![{\displaystyle \mathbf {F} =\langle 1,2,3\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae64933db3c226c9af2a7843fd90207bab5b186e)
is applied to the wrench
n units away from the origin.
, so the torque is directed along ![{\displaystyle \pm \langle 0,-3,2\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7306b673bde05bed00f33434f3676fe255604a47)
, so the torque is directed along ![{\displaystyle \pm \langle 0,-3,2\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7306b673bde05bed00f33434f3676fe255604a47)
Prove the following identities or show them false by giving a counterexample.
89.
![{\displaystyle (\mathbf {u} -\mathbf {v} )\times (\mathbf {u} +\mathbf {v} )=2(\mathbf {u} \times \mathbf {v} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d36048ff1ad3b65051034f3c94ef22aadc475f3)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle \begin{eqnarray}(\mathbf u-\mathbf v)\times(\mathbf u+\mathbf v)&=&(\mathbf u-\mathbf v)\times\mathbf u + (\mathbf u-\mathbf v)\times\mathbf v\\&=&\mathbf u\times\mathbf u - \mathbf v\times\mathbf u + \mathbf u\times\mathbf v - \mathbf v\times\mathbf v\\&=&\mathbf u\times\mathbf v-\mathbf v\times\mathbf u\\&=&\mathbf u\times\mathbf v + \mathbf u \times \mathbf v\\&=&2(\mathbf u\times\mathbf v)\end{eqnarray}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle \begin{eqnarray}(\mathbf u-\mathbf v)\times(\mathbf u+\mathbf v)&=&(\mathbf u-\mathbf v)\times\mathbf u + (\mathbf u-\mathbf v)\times\mathbf v\\&=&\mathbf u\times\mathbf u - \mathbf v\times\mathbf u + \mathbf u\times\mathbf v - \mathbf v\times\mathbf v\\&=&\mathbf u\times\mathbf v-\mathbf v\times\mathbf u\\&=&\mathbf u\times\mathbf v + \mathbf u \times \mathbf v\\&=&2(\mathbf u\times\mathbf v)\end{eqnarray}}
100. Differentiate
![{\displaystyle \mathbf {r} (t)=\langle te^{-t},t\ln t,t\cos(t)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/17314a7c81efe26c5e82f9a1b648eecc13b143b4)
.
![{\displaystyle \langle e^{-t}-te^{-t},\ln(t)+1,cos(t)-t\sin(t)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a493e2b5fddcf28013a233248e55dfca4066697b)
![{\displaystyle \langle e^{-t}-te^{-t},\ln(t)+1,cos(t)-t\sin(t)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a493e2b5fddcf28013a233248e55dfca4066697b)
101. Find a tangent vector for the curve
![{\displaystyle \mathbf {r} (t)=\langle 2t^{4},6t^{3/2},10/t\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d58da880ed1d898cf6293bac492a86129fc61e95)
at the point
![{\displaystyle t=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/970dea4a5f5ec5355c4cdd62f6396fbc8b1baaa1)
.
![{\displaystyle \langle 8,9,-10\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/13281b83b602ade75d28c791ee340a7f93817067)
![{\displaystyle \langle 8,9,-10\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/13281b83b602ade75d28c791ee340a7f93817067)
102. Find the unit tangent vector for the curve
![{\displaystyle \mathbf {r} (t)=\langle t,2,2/t\rangle ,\ t\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55e23189681f86b836c11ef60d9c34e76366950a)
.
![{\displaystyle \displaystyle {\frac {\langle t^{2},0,-2\rangle }{\sqrt {t^{4}+4}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca25deb6d0707e125a167eade15b565c33c7ca86)
![{\displaystyle \displaystyle {\frac {\langle t^{2},0,-2\rangle }{\sqrt {t^{4}+4}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca25deb6d0707e125a167eade15b565c33c7ca86)
103. Find the unit tangent vector for the curve
![{\displaystyle \mathbf {r} (t)=\langle \sin(t),\cos(t),e^{-t}\rangle ,\ t\in [0,\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2f30552af99a8f55083220a2960a8a6697bad79)
at the point
![{\displaystyle t=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c)
.
![{\displaystyle \displaystyle {\frac {\langle 1,0,-1\rangle }{\sqrt {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e18b4b877446562ee4a7549003e149aa54766c3)
![{\displaystyle \displaystyle {\frac {\langle 1,0,-1\rangle }{\sqrt {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e18b4b877446562ee4a7549003e149aa54766c3)
104. Find
![{\displaystyle \mathbf {r} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1)
if
![{\displaystyle \mathbf {r} '(t)=\langle {\sqrt {t}},\cos(\pi t),4/t\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c315146fae77e7ef4859b40635eb80771161efb9)
and
![{\displaystyle \mathbf {r} (1)=\langle 2,3,4\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a62fedc15f45ae9d0880817374bf1cd8914d3f99)
.
![{\displaystyle \displaystyle \left\langle {\frac {2t^{3/2}+4}{3}},{\frac {\sin(\pi t)}{\pi }}+3,4\ln |t|+4\right\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e18104c8338b7d633d2e9f7e0d0008fa52d80ec1)
![{\displaystyle \displaystyle \left\langle {\frac {2t^{3/2}+4}{3}},{\frac {\sin(\pi t)}{\pi }}+3,4\ln |t|+4\right\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e18104c8338b7d633d2e9f7e0d0008fa52d80ec1)
120. Find velocity, speed, and acceleration of an object if the position is given by
![{\displaystyle \mathbf {r} (t)=\langle 3\sin(t),5\cos(t),4\sin(t)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/85a4e8d4f758878d2c7cf81dd09d71840a47f59e)
.
,
, ![{\displaystyle \mathbf {a} =\langle -3\sin(t),-5\cos(t),-4\sin(t)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6d6ed7acbb6709b935b4873a4e3e28ca912d7c6)
,
, ![{\displaystyle \mathbf {a} =\langle -3\sin(t),-5\cos(t),-4\sin(t)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6d6ed7acbb6709b935b4873a4e3e28ca912d7c6)
121. Find the velocity and the position vectors for
![{\displaystyle t\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/248525429e9cd266f53ab8c52d17bc206c546060)
if the acceleration is given by
![{\displaystyle \mathbf {a} (t)=\langle e^{-t},1\rangle ,\ \mathbf {v} (0)=\langle 1,0\rangle ,\ \mathbf {r} (0)=\langle 0,0\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/60d29ee4e6618c95f5ecdca838474e8f6c0d2607)
.
, ![{\displaystyle \mathbf {r} (t)=\langle e^{-t}+2t-1,t^{2}/2\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/def7ffa6a750585493061441d347f5188734af9d)
, ![{\displaystyle \mathbf {r} (t)=\langle e^{-t}+2t-1,t^{2}/2\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/def7ffa6a750585493061441d347f5188734af9d)
Find the length of the following curves.
142. Find a description of the curve that uses arc length as a parameter:
![{\displaystyle \mathbf {r} (t)=\langle t^{2},2t^{2},4t^{2}\rangle \ t\in [1,4].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9715ec537d31fdc34e2ac352df9235bfa517bc28)
![{\displaystyle \displaystyle \mathbf {r} (s)=\left({\frac {s}{\sqrt {21}}}+1\right)\langle 1,2,4\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e99391b12706ea35251d2bfb9592712b75ee995f)
![{\displaystyle \displaystyle \mathbf {r} (s)=\left({\frac {s}{\sqrt {21}}}+1\right)\langle 1,2,4\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e99391b12706ea35251d2bfb9592712b75ee995f)
143. Find the unit tangent vector
T and the principal unit normal vector
N for the curve
![{\displaystyle \mathbf {r} (t)=\langle t^{2},t\rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3689f66f5ffe1373715b1c288720ee58e03bec63)
Check that
T⋅
N=0.
![{\displaystyle \mathbf {T} (t)=\displaystyle {\frac {\langle 2t,1\rangle }{\sqrt {4t^{2}+1}}},\ \mathbf {N} (t)=\displaystyle {\frac {\langle 1,-2t\rangle }{\sqrt {4t^{2}+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/408d07c67e1bca39fb0ec2172fa2577ff2c8c93a)
![{\displaystyle \mathbf {T} (t)=\displaystyle {\frac {\langle 2t,1\rangle }{\sqrt {4t^{2}+1}}},\ \mathbf {N} (t)=\displaystyle {\frac {\langle 1,-2t\rangle }{\sqrt {4t^{2}+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/408d07c67e1bca39fb0ec2172fa2577ff2c8c93a)
160. Find an equation of a plane passing through points
![{\displaystyle (1,1,2),\ (1,2,2),\ (-1,0,1).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0521801e98c7130ffc5dc1e6682f8ed4e81d564d)
![{\displaystyle x-2z+3=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e80655b9c17147ecf2eb3915f2f2ef510b5351ed)
![{\displaystyle x-2z+3=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e80655b9c17147ecf2eb3915f2f2ef510b5351ed)
161. Find an equation of a plane parallel to the plane 2x−y+z=1 passing through the point (0,2,-2)
![{\displaystyle 2x-y+z+4=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12ce3edd334e9e68d3d947ae55fada6775938359)
![{\displaystyle 2x-y+z+4=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12ce3edd334e9e68d3d947ae55fada6775938359)
162. Find an equation of the line perpendicular to the plane x+y+2z=4 passing through the point (5,5,5).
![{\displaystyle \mathbf {r} (t)=\langle 5+t,5+t,5+2t\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d649217d5ac3a8a47784fffd908caefb225eb21)
![{\displaystyle \mathbf {r} (t)=\langle 5+t,5+t,5+2t\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d649217d5ac3a8a47784fffd908caefb225eb21)
163. Find an equation of the line where planes x+2y−z=1 and x+y+z=1 intersect.
![{\displaystyle \mathbf {r} (t)=\langle 1-3t,2t,t\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0daae9daeeb97ef75f439fda844044bab2fe929f)
![{\displaystyle \mathbf {r} (t)=\langle 1-3t,2t,t\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0daae9daeeb97ef75f439fda844044bab2fe929f)
164. Find the angle between the planes x+2y−z=1 and x+y+z=1.
![{\displaystyle \cos ^{-1}{\frac {2}{\sqrt {18}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92ca84f6ed35a4a64eea1ec65fc65d3d111dd551)
![{\displaystyle \cos ^{-1}{\frac {2}{\sqrt {18}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92ca84f6ed35a4a64eea1ec65fc65d3d111dd551)
165. Find the distance from the point (3,4,5) to the plane x+y+z=1.
![{\displaystyle {\frac {11}{3}}{\sqrt {3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e495805488aff299d937b02379974c210cc6f0a7)
![{\displaystyle {\frac {11}{3}}{\sqrt {3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e495805488aff299d937b02379974c210cc6f0a7)
Evaluate the following limits.
180.
![{\displaystyle \displaystyle \lim _{(x,y)\rightarrow (1,-2)}{\frac {y^{2}+2xy}{y+2x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebcb684f07cfbcf3df20297f75d0bb3fe0179fb5)
181.
![{\displaystyle \displaystyle \lim _{(x,y)\rightarrow (4,5)}{\frac {{\sqrt {x+y}}-3}{x+y-9}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c9d03ed5480a6d5c2799367ccb412cdda00bcb6)
At what points is the function f continuous?
183.
![{\displaystyle f(x,y)=\displaystyle {\frac {\ln(x^{2}+y^{2})}{x-y+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67eb097c7031711caf5807cb61be6827e4c211b8)
All points (x,y) except for (0,0) and the line y=x+1
All points (x,y) except for (0,0) and the line y=x+1
Use the two-path test to show that the following limits do not exist. (A path does not have to be a straight line.)
184.
![{\displaystyle \displaystyle \lim _{(x,y)\rightarrow (0,0)}{\frac {4xy}{3x^{2}+y^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0bf345f40643afd561dcdcbc668e1a58f7f0e6b)
The limit is 1 along the line y=x, and −1 along the line y=−x
The limit is 1 along the line y=x, and −1 along the line y=−x
186.
![{\displaystyle \displaystyle \lim _{(x,y)\rightarrow (0,0)}{\frac {x^{3}-y^{2}}{x^{3}+y^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96a2e399deb0b86fc6d3c12e420ee28ced53963c)
The limit is 1 along the line y=0, and −1 along the line x=0
The limit is 1 along the line y=0, and −1 along the line x=0
201. Find all three partial derivatives of the function
![{\displaystyle \displaystyle f(x,y,z)=xe^{y^{2}+z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2161eebe16dc5f0baa34fbfe03d9e5694ec9c224)
![{\displaystyle \displaystyle f_{x}=e^{y^{2}+z},\ f_{y}=2xye^{y^{2}+z},\ f_{z}=f.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5683300bea5c2e6e1d4779ad4baa4105feafdaf)
![{\displaystyle \displaystyle f_{x}=e^{y^{2}+z},\ f_{y}=2xye^{y^{2}+z},\ f_{z}=f.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5683300bea5c2e6e1d4779ad4baa4105feafdaf)
Find the four second partial derivatives of the following functions.
202.
![{\displaystyle f(x,y)=\cos(xy)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba1d272176c97cbb1fbeea4554140a878c69657a)
![{\displaystyle f_{xx}=-y^{2}\cos(xy),\ f_{yy}=-x^{2}\cos(xy),\ f_{xy}=f_{yx}=-\sin(xy)-xy\cos(xy).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e426082af21f5a43f87b7bcaaa1766c3569332c8)
![{\displaystyle f_{xx}=-y^{2}\cos(xy),\ f_{yy}=-x^{2}\cos(xy),\ f_{xy}=f_{yx}=-\sin(xy)-xy\cos(xy).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e426082af21f5a43f87b7bcaaa1766c3569332c8)
203.
![{\displaystyle f(x,y)=xe^{y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6cc1e77a2278e0b5e037b2279dc919f171f264)
![{\displaystyle f_{xx}=0,\ f_{yy}=xe^{y},\ f_{xy}=f_{yx}=e^{y}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9045f47bf330743347267e6c360656a8b4ba8eb)
![{\displaystyle f_{xx}=0,\ f_{yy}=xe^{y},\ f_{xy}=f_{yx}=e^{y}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9045f47bf330743347267e6c360656a8b4ba8eb)
Find
221.
![{\displaystyle f(x,y)={\sqrt {x^{2}+y^{2}}},\ x(t)=\cos(2t),\ y(t)=\sin(2t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b75f46b32b6976789d4d1b3dee6c2f38478769b9)
222.
![{\displaystyle \displaystyle f(x,y,z)={\frac {x-y}{y+z}},\ x(t)=t,\ \displaystyle y(t)=2t,\ z(t)=3t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1907538ad615ebbb0f65839249d79cec414a5611)
Find
223.
![{\displaystyle f(x,y)=\sin(x)\cos(2y),\ x=s+t,\ y=s-t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc83b657165c2155f0231dde30d50a440a613474)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle f_s = \cos(s+t)\cos(2s-2t) - 2\sin(s+t)\sin(2s-2t)\\ f_t = \cos(s+t)\cos(2s-2t) + 2\sin(s+t)\sin(2s-2t)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle f_s = \cos(s+t)\cos(2s-2t) - 2\sin(s+t)\sin(2s-2t)\\ f_t = \cos(s+t)\cos(2s-2t) + 2\sin(s+t)\sin(2s-2t)}
224.
![{\displaystyle \displaystyle f(x,y,z)={\frac {x-z}{y+z}},\ x(t)=s+t,\ y(t)=st,\ z(t)=s-t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/283e90dd2ce5331b454967d79bef102af7a37748)
Failed to parse (syntax error): {\displaystyle \displaystyle f_s = \frac{-2t(t+1)}{(st+s-t)^2}\\ \displaystyle f_t = \frac{2s}{(st+s-t)^2}}
Failed to parse (syntax error): {\displaystyle \displaystyle f_s = \frac{-2t(t+1)}{(st+s-t)^2}\\ \displaystyle f_t = \frac{2s}{(st+s-t)^2}}
225. The volume of a pyramid with a square base is
![{\displaystyle V={\frac {1}{3}}x^{2}h}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de9ce6ad029040bf3134c9692d0af40f5b4f2a25)
, where
x is the side of the square base and
h is the height of the pyramid. Suppose that
![{\displaystyle \displaystyle x(t)={\frac {t}{t+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27e535cd1991ddf259ea8b9985f08322c268f825)
and
![{\displaystyle \displaystyle h(t)={\frac {1}{t+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bc7f9494c057e079c6f262a20f84cb7757d04f)
for
![{\displaystyle t\geq 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb260e03f8f2bf09d20a8369fb31b3988501014)
Find
![{\displaystyle V'(t).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3002031ba583aa07c59da516f16e188bcd3a5602)
![{\displaystyle \displaystyle {\frac {2t-t^{2}}{3(t+1)^{4}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2957372b23cb5d9028b3ee1c901df85d7f6581)
![{\displaystyle \displaystyle {\frac {2t-t^{2}}{3(t+1)^{4}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2957372b23cb5d9028b3ee1c901df85d7f6581)
Find an equation of a plane tangent to the given surface at the given point(s).
240.
![{\displaystyle xy\sin(z)=1,\ (1,2,\pi /6),\ (-1,-2,5\pi /6).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/114a1efaa3ea6778f3235a323753f3f335683488)
![{\displaystyle (x-1)+{\frac {1}{2}}(y-2)+{\sqrt {3}}(z-\pi /6)=0,\ (x+1)+{\frac {1}{2}}(y+2)+{\sqrt {3}}(z-5\pi /6)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae0ee90e1005cb3fa520f040d9ef00e908fb2b8)
![{\displaystyle (x-1)+{\frac {1}{2}}(y-2)+{\sqrt {3}}(z-\pi /6)=0,\ (x+1)+{\frac {1}{2}}(y+2)+{\sqrt {3}}(z-5\pi /6)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae0ee90e1005cb3fa520f040d9ef00e908fb2b8)
241.
![{\displaystyle z=x^{2}e^{x-y},\ (2,2,4),\ (-1,-1,1).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a39ab5e2b06a6e0bb1bb5d800d8b353412165f6f)
![{\displaystyle -8(x-2)+4(y-2)+z-4=0,\ x+y+z+1=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc74a7f63a4e97b73e815342f8d2247d3839ecc)
![{\displaystyle -8(x-2)+4(y-2)+z-4=0,\ x+y+z+1=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc74a7f63a4e97b73e815342f8d2247d3839ecc)
Find critical points of the function f. When possible, determine whether each critical point corresponds to a local maximum, a local minimum, or a saddle point.
260.
![{\displaystyle f(x,y)=x^{4}+2y^{2}-4xy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a1c445e7214a6b2d924712c498a15a07d6dba2)
Local minima at (1,1) and (−1,−1), saddle at (0,0)
Local minima at (1,1) and (−1,−1), saddle at (0,0)
261.
![{\displaystyle f(x,y)=\tan ^{-1}(xy)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e84fbb10c429427caca59ef04d46bb2bdf3607)
262.
![{\displaystyle f(x,y)=2xye^{-x^{2}-y^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59076ebe29ea207c890ee9a748b89ba48db22315)
Saddle at (0,0), local maxima at
local minima at ![{\displaystyle (\pm 1/{\sqrt {2}},\mp 1/{\sqrt {2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/177e9620f311a421e11d89435014d73605eebddb)
Saddle at (0,0), local maxima at
local minima at ![{\displaystyle (\pm 1/{\sqrt {2}},\mp 1/{\sqrt {2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/177e9620f311a421e11d89435014d73605eebddb)
Find absolute maximum and minimum values of the function f on the set R.
263.
![{\displaystyle f(x,y)=x^{2}+y^{2}-2y+1,\ R=\{(x,y)\mid x^{2}+y^{2}\leq 4\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25fa736e06e7208a73dd247973bc3096f95d102b)
Maximum of 9 at (0,−2) and minimum of 0 at (0,1)
Maximum of 9 at (0,−2) and minimum of 0 at (0,1)
264.
R is a closed triangle with vertices (0,0), (2,0), and (0,2).
Maximum of 0 at (2,0), (0,2), and (0,0); minimum of −2 at (1,1)
Maximum of 0 at (2,0), (0,2), and (0,0); minimum of −2 at (1,1)
265. Find the point on the plane x−y+z=2 closest to the point (1,1,1).
![{\displaystyle (4/3,2/3,4/3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a61ed8c63841bee2b28adcb159041bbb4b458c5)
![{\displaystyle (4/3,2/3,4/3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a61ed8c63841bee2b28adcb159041bbb4b458c5)
Evaluate the given integral over the region R.
280.
![{\displaystyle \displaystyle \iint _{R}(x^{2}+xy)dA,\ R=\{(x,y)\mid x\in [1,2],\ y\in [-1,1]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed365e6e7ddc55551e813614af01d9c10cdc9af6)
![{\displaystyle 14/3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5aba8b23625eb49fef65b0a948730453e5e427b7)
![{\displaystyle 14/3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5aba8b23625eb49fef65b0a948730453e5e427b7)
281.
![{\displaystyle \displaystyle \iint _{R}(xy\sin(x^{2}))dA,\ R=\{(x,y)\mid x\in [0,{\sqrt {\pi /2}}],\ y\in [0,1]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/348d8c966096be3b39bffd752544bc134cb87182)
![{\displaystyle 1/4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3cf1ef33695c3d98cb09f01e5700f927ce928c)
![{\displaystyle 1/4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3cf1ef33695c3d98cb09f01e5700f927ce928c)
282.
![{\displaystyle \displaystyle \iint _{R}{\frac {x}{(1+xy)^{2}}}dA,\ R=\{(x,y)\mid x\in [0,4],\ y\in [1,2]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d6288e4ccdb0134f6d2546fdd60591316f57ca)
![{\displaystyle \ln(5/3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76ec600eb74181b6432e1121b6534b74634bb1a2)
![{\displaystyle \ln(5/3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76ec600eb74181b6432e1121b6534b74634bb1a2)
Evaluate the given iterated integrals.
Evaluate the following integrals.
300.
R is bounded by
x=0,
y=2
x+1, and
y=5−2
x.
![{\displaystyle 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f)
![{\displaystyle 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f)
301.
R is in the first quadrant and bounded by
x=0,
![{\displaystyle y=x^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e06a3fb2a27eccbf6e31de4fa547f10784df851)
and
![{\displaystyle y=8-x^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/648c65b0720e93c075f8c6eda6a322737362d694)
![{\displaystyle 152/3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e93d05b5156a3fbbf4e91b395ed5560eef2eb1f1)
![{\displaystyle 152/3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e93d05b5156a3fbbf4e91b395ed5560eef2eb1f1)
Use double integrals to compute the volume of the given region.
323. Evaluate
![{\displaystyle \displaystyle \iint _{R}{\frac {x-y}{x^{2}+y^{2}+1}}dA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21709c9db522102a88cb6b15af60ef48a570bb6a)
if
R is the unit disk centered at the origin.
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
In the following exercises, sketching the region of integration may be helpful.
341. Find the volume of the solid in the first octant bounded by the plane 2x+3y+6z=12 and the coordinate planes.
![{\displaystyle 8}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1aaa997e6ad67716cfaa9a02c4df860bf60a95b5)
![{\displaystyle 8}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1aaa997e6ad67716cfaa9a02c4df860bf60a95b5)
342. Find the volume of the solid in the first octant bounded by the cylinder
![{\displaystyle z=\sin(y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afb4c3adbe65cfc1e0dd92a9d6cb5cf9a95090e3)
for
![{\displaystyle y\in [0,\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0281fdf747a9247ed8a56221ddd07b34c2a50097)
, and the planes
y=
x and
x=0.
![{\displaystyle \pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a)
![{\displaystyle \pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a)
344. Rewrite the integral
![{\displaystyle \displaystyle \int _{0}^{1}\int _{-2}^{2}\int _{0}^{\sqrt {4-y^{2}}}dzdydx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7867366b6580949fa9cd0997759140d0baf564e)
in the order
dydzdx.
![{\displaystyle \displaystyle \int _{0}^{1}\int _{0}^{2}\int _{-{\sqrt {4-z^{2}}}}^{\sqrt {4-z^{2}}}dydzdx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49c8c7ac2b5595f94dad5e896319604344321768)
![{\displaystyle \displaystyle \int _{0}^{1}\int _{0}^{2}\int _{-{\sqrt {4-z^{2}}}}^{\sqrt {4-z^{2}}}dydzdx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49c8c7ac2b5595f94dad5e896319604344321768)
361. Find the mass of the solid cylinder
![{\displaystyle D=\{(r,\theta ,z)\mid r\in [0,3],\ z\in [0,2]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3d5655e4f2397642509c6640400d7087e95cf35)
given the density function
![{\displaystyle \delta (r,\theta ,z)=5e^{-r^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae1f6834555d1dd000077a0627aefbf6d5d234ed)
![{\displaystyle 10\pi (1-e^{-9})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd866b11f9b4e7a51ab5791e303533a363471458)
![{\displaystyle 10\pi (1-e^{-9})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd866b11f9b4e7a51ab5791e303533a363471458)
362. Use a triple integral to find the volume of the region bounded by the plane
z=0 and the hyperboloid
![{\displaystyle z={\sqrt {17}}-{\sqrt {1+x^{2}+y^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd2458a938a5202f933c5e456ce9a26945661051)
![{\displaystyle \displaystyle {\frac {2\pi (1+7{\sqrt {17}})}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe8c4357ba9fef651470f9cfc45c0abca8313973)
![{\displaystyle \displaystyle {\frac {2\pi (1+7{\sqrt {17}})}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe8c4357ba9fef651470f9cfc45c0abca8313973)
363. If
D is a unit ball, use a triple integral in spherical coordinates to evaluate
![{\displaystyle \iiint _{D}(x^{2}+y^{2}+z^{2})^{5/2}dV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/223a2baa893cb046b86b90b67460890b83810ff1)
![{\displaystyle \pi /2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44e3d874a0b229fded7ffce67a0677dd5b8b67)
![{\displaystyle \pi /2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44e3d874a0b229fded7ffce67a0677dd5b8b67)
364. Find the mass of a solid cone
![{\displaystyle \{(\rho ,\phi ,\theta )\mid \phi \leq \pi /3,\ z\in [0,4]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f200079667c56d8b47444f8062308745eb385015)
if the density function is
![{\displaystyle \delta (\rho ,\phi ,\theta )=5-z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9478e110db7bc776f218b7884925de3afa3525a)
![{\displaystyle 128\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f3f1bdb494819c8fbaa95c07293525b9581e74)
![{\displaystyle 128\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f3f1bdb494819c8fbaa95c07293525b9581e74)
380. Find the center of mass for three particles located in space at (1,2,3), (0,0,1), and (1,1,0), with masses 2, 1, and 1 respectively.
![{\displaystyle {\frac {\langle 3,5,7\rangle }{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6332a924ef6c692c318b32da86a2c06ebbadfddb)
![{\displaystyle {\frac {\langle 3,5,7\rangle }{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6332a924ef6c692c318b32da86a2c06ebbadfddb)
384. Find the centroid of the region in the first quadrant bounded by
![{\displaystyle y=\ln(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6a4b4bf3980aaa7227cb1fca33d9864a8bb4b47)
,
![{\displaystyle y=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/094f824655138f6b11d96a0da32e7f0716ba6959)
, and
![{\displaystyle x=e}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba47ab1931fc4886c5da08831962cc141d20655)
.
![{\displaystyle ((e^{2}+1)/4,e/2-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e980929a84e77eb3a1afc014870be9172b909595)
![{\displaystyle ((e^{2}+1)/4,e/2-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e980929a84e77eb3a1afc014870be9172b909595)
385. Find the center of mass for the region
![{\displaystyle \{(x,y)\mid x\in [0,4],y\in [0,2]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e476dabc9b7a7dfe36cd98372488912496465f7)
, with the density
![{\displaystyle \rho (x,y)=1+x/2.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96618c2fa7107b7b8da2fa03cdbf6642860fecbf)
![{\displaystyle (7/3,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/289e8abbf05e509e7826c279bf13227bffb0f31d)
![{\displaystyle (7/3,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/289e8abbf05e509e7826c279bf13227bffb0f31d)
386. Find the center of mass for the triangular plate with vertices (0,0), (0,4), and (4,0), with density
![{\displaystyle \rho (x,y)=1+x+y.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4d666834cb75e6ac4e82dacb1b584a509ea7d2b)
![{\displaystyle (16/11,16/11)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00fa85c5e62696d2809fe8ea75ba6225c46da1ae)
![{\displaystyle (16/11,16/11)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00fa85c5e62696d2809fe8ea75ba6225c46da1ae)
One can sketch two-dimensional vector fields by plotting vector values, flow curves, and/or equipotential curves.
402. Find and sketch the gradient field
![{\displaystyle \mathbf {F} =\nabla \phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/24c10e4f746d8ab5226bb1db9dacda8b185a89af)
for the potential function
![{\displaystyle \phi (x,y)=\sin(x)\sin(y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0fcec58b1d9a99ce189585809c8acb4c338280)
for
![{\displaystyle |x|\leq \pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c1f1f5476d6277e581c15192fe9c5312db24dc1)
and
![{\displaystyle |y|\leq \pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/813925ba0fc80d16f44eaa977f167cf8b2e45b8d)
.
![{\displaystyle \nabla \phi (x,y)=\langle \cos(x)\sin(y),\sin(x)\cos(y)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f67b952e709d7c5bdd016b8b1d062cb8f935c01)
![{\displaystyle \nabla \phi (x,y)=\langle \cos(x)\sin(y),\sin(x)\cos(y)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f67b952e709d7c5bdd016b8b1d062cb8f935c01)
403. Find the gradient field
![{\displaystyle \mathbf {F} =\nabla \phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/24c10e4f746d8ab5226bb1db9dacda8b185a89af)
for the potential function
![{\displaystyle \phi (x,y,z)=e^{-z}\sin(x+y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bc8e16d40c5f7418c908c1de0dbe618a11d3e20)
![{\displaystyle \mathbf {F} =e^{-z}\left\langle \cos(x+y),\cos(x+y),-\sin(x+y)\right\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0055fc3a8578eb4ba273f863b98306b8f388ca8)
![{\displaystyle \mathbf {F} =e^{-z}\left\langle \cos(x+y),\cos(x+y),-\sin(x+y)\right\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0055fc3a8578eb4ba273f863b98306b8f388ca8)
420. Evaluate
![{\displaystyle \int _{C}(x^{2}+y^{2})ds}](https://wikimedia.org/api/rest_v1/media/math/render/svg/703c80b06911ba8c668f7a7b65ad254860e19f12)
if
C is the line segment from (0,0) to (5,5)
![{\displaystyle {\frac {250{\sqrt {2}}}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba575a2afb84a6aae5c4ad2a926af0cf3b0d2fb0)
![{\displaystyle {\frac {250{\sqrt {2}}}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba575a2afb84a6aae5c4ad2a926af0cf3b0d2fb0)
421. Evaluate
![{\displaystyle \int _{C}(x^{2}+y^{2})ds}](https://wikimedia.org/api/rest_v1/media/math/render/svg/703c80b06911ba8c668f7a7b65ad254860e19f12)
if
C is the circle of radius 4 centered at the origin
![{\displaystyle 128\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f3f1bdb494819c8fbaa95c07293525b9581e74)
![{\displaystyle 128\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f3f1bdb494819c8fbaa95c07293525b9581e74)
422. Evaluate
![{\displaystyle \int _{C}(y-z)ds}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd22650c4db2c96bb56fc346feec15731fad2f85)
if
C is the helix
![{\displaystyle \mathbf {r} (t)=\langle 3\cos(t),3\sin(t),t\rangle ,\ t\in [0,2\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c08e2103ca08229383eb122b8d4d0b8dba3eac9)
![{\displaystyle -2{\sqrt {10}}\pi ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0854611bc680149cac184dd7252ba9cd61ced7c)
![{\displaystyle -2{\sqrt {10}}\pi ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0854611bc680149cac184dd7252ba9cd61ced7c)
423. Evaluate
![{\displaystyle \int _{C}\mathbf {F} \cdot d\mathbf {r} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d972b821eeed2f0528814bd37e9a210592b600f1)
if
![{\displaystyle \mathbf {F} =\langle x,y\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a961e1a2b386f164460acb5da6152e81216d28d)
and
C is the arc of the parabola
![{\displaystyle \mathbf {r} (t)=\langle 4t,t^{2}\rangle ,\ t\in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/642cd051fef71427b9ff15e0885d8b70e1d1f344)
![{\displaystyle 17/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2ab81608c89542f38877067d909f02d008d232)
![{\displaystyle 17/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2ab81608c89542f38877067d909f02d008d232)
Determine if the following vector fields are conservative on
440.
![{\displaystyle \langle -y,x+y\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5396577d51124348a0a78d7b9bf1808009db967)
441.
![{\displaystyle \langle 2x^{3}+xy^{2},2y^{3}+x^{2}y\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/247696a59731cadb50bde2f7a2ad65b101bb379b)
Determine if the following vector fields are conservative on their respective domains in
When possible, find the potential function.
442.
![{\displaystyle \langle y,x,1\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/95c3f194a95b2fc175ee2a7c37b5d627059dbbfa)
![{\displaystyle \phi (x,y,z)=xy+z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9029f2e3b55128f6b9656e68cc3ea1cd560363de)
![{\displaystyle \phi (x,y,z)=xy+z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9029f2e3b55128f6b9656e68cc3ea1cd560363de)
443.
![{\displaystyle \langle x^{3},2y,-z^{3}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b81e95e08ec308c8593bf3af6880fe7771e190d6)
![{\displaystyle \phi (x,y,z)=(x^{4}+4y^{2}-z^{4})/4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3be02f5b735e399b4548728942eb0f536f021e4b)
![{\displaystyle \phi (x,y,z)=(x^{4}+4y^{2}-z^{4})/4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3be02f5b735e399b4548728942eb0f536f021e4b)
460. Evaluate the circulation of the field
![{\displaystyle \mathbf {F} =\langle 2xy,x^{2}-y^{2}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cec48bf9a4e2233fa0b67f33d229f55a4e5de9d8)
over the boundary of the region above
y=0 and below
y=
x(2-
x) in two different ways, and compare the answers.
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
461. Evaluate the circulation of the field
![{\displaystyle \mathbf {F} =\langle 0,x^{2}+y^{2}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/309e8c3600bf24b333876f9533939b626bc1fb78)
over the unit circle centered at the origin in two different ways, and compare the answers.
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
462. Evaluate the flux of the field
![{\displaystyle \mathbf {F} =\langle y,-x\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5b344898e27a173253259fb9788bb6ea91f16c4)
over the square with vertices (0,0), (1,0), (1,1), and (0,1) in two different ways, and compare the answers.
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
482. Find the curl of
![{\displaystyle \langle x^{2}-y^{2},xy,z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/201e6047a702995a6baa27da278d38652fcc0a0d)
![{\displaystyle \langle 0,0,3y\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cac2a763560ae0e692e5e8b336f10463741dea07)
![{\displaystyle \langle 0,0,3y\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cac2a763560ae0e692e5e8b336f10463741dea07)
484. Prove that the general rotation field
![{\displaystyle \mathbf {F} =\mathbf {a} \times \mathbf {r} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bf675ad6ca689142c4864d3e841979e2da5761)
, where
![{\displaystyle \mathbf {a} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d)
is a non-zero constant vector and
![{\displaystyle \mathbf {r} =\langle x,y,z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ead22d37bd8c3bd1bede7f57f519eb6845e52eeb)
, has zero divergence, and the curl of
![{\displaystyle \mathbf {F} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/da18bef8c979f3548bb0d8976f5844012d7b8256)
is
![{\displaystyle 2\mathbf {a} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/47844954e772249023020d9a1ffe5235551b76a1)
.
If
, then
, and then
![{\displaystyle \nabla \times \mathbf {F} =\langle h_{y}-g_{z},f_{z}-h_{x},g_{x}-f_{y}\rangle =\langle a_{1}+a_{1},a_{2}+a_{2},a_{3}+a_{3}\rangle =2\mathbf {a} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19a80883b31c0670c839a9bb9a142c9e8204da46)
If
, then
, and then
![{\displaystyle \nabla \times \mathbf {F} =\langle h_{y}-g_{z},f_{z}-h_{x},g_{x}-f_{y}\rangle =\langle a_{1}+a_{1},a_{2}+a_{2},a_{3}+a_{3}\rangle =2\mathbf {a} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19a80883b31c0670c839a9bb9a142c9e8204da46)
500. Give a parametric description of the plane
![{\displaystyle 2x-4y+3z=16.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a87717df129166d0da156dd69fe8bfa95afa325c)
![{\displaystyle \langle u,v,(16-2u+4v)/3\rangle ,\ u,v\in \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c129ca25e6fc4fa0e072e4218dba618f6cadac0)
![{\displaystyle \langle u,v,(16-2u+4v)/3\rangle ,\ u,v\in \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c129ca25e6fc4fa0e072e4218dba618f6cadac0)
501. Give a parametric description of the hyperboloid
![{\displaystyle z^{2}=1+x^{2}+y^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36ce9240c728e0b1c15d44787db78f723b003b80)
![{\displaystyle \langle {\sqrt {v^{2}-1}}\cos(u),{\sqrt {v^{2}-1}}\sin(u),v\rangle ,\ u\in [0,2\pi ],\ |v|\geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e30ae2582ed79c8841a5ec44ce9a38adb627be5)
![{\displaystyle \langle {\sqrt {v^{2}-1}}\cos(u),{\sqrt {v^{2}-1}}\sin(u),v\rangle ,\ u\in [0,2\pi ],\ |v|\geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e30ae2582ed79c8841a5ec44ce9a38adb627be5)
502. Integrate
![{\displaystyle f(x,y,z)=xy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f27606abbc927475726d0fdce8b33b8b7cce268)
over the portion of the plane
z=2−
x−
y in the first octant.
![{\displaystyle 2/{\sqrt {3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eb0bea2fe9088875244141adf992b51e8fa12c3)
![{\displaystyle 2/{\sqrt {3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eb0bea2fe9088875244141adf992b51e8fa12c3)
504. Find the flux of the field
![{\displaystyle \mathbf {F} =\langle x,y,z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f902ca1df7a2a32a0451a7fa029cbedcb16b911)
across the surface of the cone
![{\displaystyle z^{2}=x^{2}+y^{2},\ z\in [0,1],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdef9c8592f3f4001cfc49e3df0f914ece0b10df)
with normal vectors pointing in the positive
z direction.
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
505. Find the flux of the field
![{\displaystyle \mathbf {F} =\langle -y,z,1\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e33820b226c94ee4cc6192525a12b5c353d225a5)
across the surface
![{\displaystyle y=x^{2},\ z\in [0,4],\ x\in [0,1],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75190cd1fce58f3ae022e60f9e482de598e957cd)
with normal vectors pointing in the positive
y direction.
![{\displaystyle -10}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cae1245b26372e6d9637c98edea699eb2daaaf0b)
![{\displaystyle -10}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cae1245b26372e6d9637c98edea699eb2daaaf0b)
520. Use a surface integral to evaluate the circulation of the field
![{\displaystyle \mathbf {F} =\langle x^{2}-z^{2},y,2xz\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/439c246c646c331987b4ae83272edca755a0360e)
on the boundary of the plane
![{\displaystyle z=4-x-y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06bee1e94db0c06ce0dc9b5c1229985d620a94ce)
in the first octant.
![{\displaystyle {\frac {-128}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d0b4a8ac15057ab7d30e9a0d0dc332bfcdc6903)
![{\displaystyle {\frac {-128}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d0b4a8ac15057ab7d30e9a0d0dc332bfcdc6903)
522. Use a line integral to find
where
![{\displaystyle \mathbf {F} =\langle x,y,z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f902ca1df7a2a32a0451a7fa029cbedcb16b911)
,
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
is the upper half of the ellipsoid
![{\displaystyle {\frac {x^{2}}{4}}+{\frac {y^{2}}{9}}+z^{2}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7784a34bf21a6187e51888ab1fa67dfc3cd2840)
, and
![{\displaystyle \mathbf {n} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46)
points in the direction of the
z-axis.
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
523. Use a line integral to find
where
![{\displaystyle \mathbf {F} =\langle 2y,-z,x-y-z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b6fa35e54dffb79d7db25a7b07efd7bced82f5b)
,
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
is the part of the sphere
![{\displaystyle x^{2}+y^{2}+z^{2}=25}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75ccdca6176fc4c81546ba8ca563cd1b0826e282)
for
![{\displaystyle 3\leq z\leq 5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c6baa3584177c4b3140a5ec7224b3205258e206)
, and
![{\displaystyle \mathbf {n} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46)
points in the direction of the
z-axis.
![{\displaystyle -32\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/955d27cf435b5b3a32cbcde91325ac0188ac6143)
![{\displaystyle -32\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/955d27cf435b5b3a32cbcde91325ac0188ac6143)
Compute the net outward flux of the given field across the given surface.
540.
![{\displaystyle \mathbf {F} =\langle x,-2y,3z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b48865c18e27292e73f16aef87691e57811f222b)
,
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
is a sphere of radius
![{\displaystyle {\sqrt {6}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a857de6bca2591cfad08e4378634825b6be66a01)
centered at the origin.
![{\displaystyle 16{\sqrt {6}}\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e56e6edd17d188268c63b773bdee7d3388edb122)
![{\displaystyle 16{\sqrt {6}}\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e56e6edd17d188268c63b773bdee7d3388edb122)
542.
![{\displaystyle \mathbf {F} =\langle y+z,x+z,x+y\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4f09639d07e2dce9384f8cab5e87e2f8a9287d8)
,
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
is the boundary of the cube
![{\displaystyle \{(x,y,z)\mid |x|\leq 1,|y|\leq 1,|z|\leq 1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04cb029fb39e62d41aa1ae0064589be574fbd806)
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
543.
![{\displaystyle \mathbf {F} =\langle x,y,z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f902ca1df7a2a32a0451a7fa029cbedcb16b911)
,
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
is the surface of the region bounded by the paraboloid
![{\displaystyle z=4-x^{2}-y^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20cfd74ab05572b01e6f1909fd54396ce0406f28)
and the
xy-plane.
![{\displaystyle 24\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d591208ffb8cd424b01602f3fa652021a4015314)
![{\displaystyle 24\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d591208ffb8cd424b01602f3fa652021a4015314)
544.
![{\displaystyle \mathbf {F} =\langle z-x,x-y,2y-z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/80caeafe4262109998180fa8966c3baec8bf23b9)
,
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
is the boundary of the region between the concentric spheres of radii 2 and 4, centered at the origin.
![{\displaystyle -224\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c7f812d29f8cd9a5298efec1616ac631a890aef)
![{\displaystyle -224\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c7f812d29f8cd9a5298efec1616ac631a890aef)
545.
![{\displaystyle \mathbf {F} =\langle x,2y,3z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/63f64405ed59d8c593f1c5f58456851cb5a68765)
,
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
is the boundary of the region between the cylinders
![{\displaystyle x^{2}+y^{2}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec84b90236512e8d27ff1a8f7707b60b63327de1)
and
![{\displaystyle x^{2}+y^{2}=4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b41e02c83bfa96c6d425137298c311a50eeac17)
and cut off by planes
![{\displaystyle z=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b92bfc06485cc90286474b14a516a68d8bfdd7b3)
and
![{\displaystyle z=8}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b37480deb7a0d26a9aab0a0c8b56bd0b55199a3b)
![{\displaystyle 144\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0043fece2cb48fd081c8878c51e4fc93ad8ff94d)
![{\displaystyle 144\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0043fece2cb48fd081c8878c51e4fc93ad8ff94d)