User:Melikamp/ma225

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Parametric Equations

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

2. Find parametric equations describing the line segment from

P(x_{1},y_{1})

to

Q(x_{2},y_{2})

.

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

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

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.

x=3\cos(-t),\ y=1.5\sin(-t)

x=3\cos(-t),\ y=1.5\sin(-t)

Polar Coordinates

20. Convert the equation into Cartesian coordinates:

r=\sin(\theta )\sec ^{2}(\theta ).

y=x^{2}

y=x^{2}

21. Find an equation of the line y=mx+b in polar coordinates.

r={\frac {b}{\sin(\theta )-m\cos(\theta )}}

r={\frac {b}{\sin(\theta )-m\cos(\theta )}}

Sketch the following polar curves without using a computer.

22.

r=2-2\sin(\theta )

23.

r^{2}=4\cos(\theta )

24.

r=2\sin(5\theta )

Sketch the following sets of points.

25.

\{(r,\theta ):\theta =2\pi /3\}

26.

\{(r,\theta ):|\theta |\leq \pi /3{\mbox{ and }}|r|<3\}

Calculus in Polar Coordinates

Find points where the following curves have vertical or horizontal tangents.

40.

r=4\cos(\theta )

Horizontal tangents at (2,2) and (2,-2); vertical tangents at (0,0) and (4,0)

41.

r=2+2\sin(\theta )

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.

42. The region inside the limaçon

2+\cos(\theta )

9π/2

9π/2

43. The region inside the petals of the rose

4\cos(2\theta )

and outside the circle

r=2

8\pi /3+4{\sqrt {3}}

8\pi /3+4{\sqrt {3}}

Vectors and Dot Product

60. Find an equation of the sphere with center (1,2,0) passing through the point (3,4,5)

(x-1)^{2}+(y-2)^{2}+z^{2}=33

(x-1)^{2}+(y-2)^{2}+z^{2}=33

61. Sketch the plane passing through the points (2,0,0), (0,3,0), and (0,0,4)

62. Find the value of

|\mathbf {u} +3\mathbf {v} |

if

\mathbf {u} =\langle 1,3,0\rangle

and

\mathbf {v} =\langle 3,0,2\rangle

{\sqrt {145}}

{\sqrt {145}}

63. Find all unit vectors parallel to

\langle 1,2,3\rangle

\pm {\frac {1}{\sqrt {14}}}\langle 1,2,3\rangle

\pm {\frac {1}{\sqrt {14}}}\langle 1,2,3\rangle

64. Prove one of the distributive properties for vectors in

\mathbb {R} ^{3}

:

c(\mathbf {u} +\mathbf {v} )=c\mathbf {u} +c\mathbf {v}

Failed to parse (unknown function "\begin{eqnarray}"): {\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}}

65. Find all unit vectors orthogonal to

3\mathbf {i} +4\mathbf {j}

in

\mathbb {R} ^{2}

\pm \left\langle {\frac {-4}{5}},{\frac {3}{5}}\right\rangle

\pm \left\langle {\frac {-4}{5}},{\frac {3}{5}}\right\rangle

66. Find all unit vectors orthogonal to

3\mathbf {i} +4\mathbf {j}

in

\mathbb {R} ^{3}

\left\langle {\frac {4}{5}}c,-{\frac {3}{5}}c,{\sqrt {1-c^{2}}}\right\rangle ,\ c\in [-1,1]

\left\langle {\frac {4}{5}}c,-{\frac {3}{5}}c,{\sqrt {1-c^{2}}}\right\rangle ,\ c\in [-1,1]

67. Find all unit vectors that make an angle of

\pi /3

with the vector

\langle 1,2\rangle

{\frac {\sqrt {5}}{10}}\left\langle 1\pm 2{\sqrt {3}},\ 2\mp {\sqrt {3}}\right\rangle

{\frac {\sqrt {5}}{10}}\left\langle 1\pm 2{\sqrt {3}},\ 2\mp {\sqrt {3}}\right\rangle

Cross Product

Find $\mathbf {u} \times \mathbf {v}$ and $\mathbf {v} \times \mathbf {u}$

80.

\mathbf {u} =\langle -4,1,1\rangle

and

\mathbf {v} =\langle 0,1,-1\rangle

\mathbf {u} \times \mathbf {v} =\langle -2,-4,-4\rangle

\mathbf {u} \times \mathbf {v} =\langle -2,-4,-4\rangle

81.

\mathbf {u} =\langle 1,2,-1\rangle

and

\mathbf {v} =\langle 3,-4,6\rangle

\mathbf {u} =\langle 8,-9,-10\rangle

\mathbf {u} =\langle 8,-9,-10\rangle

Find the area of the parallelogram with sides $\mathbf {u}$ and $\mathbf {v}$ .

82.

\mathbf {u} =\langle -3,0,2\rangle

and

\mathbf {v} =\langle 1,1,1\rangle

{\sqrt {38}}

{\sqrt {38}}

83.

\mathbf {u} =\langle 8,2,-3\rangle

and

\mathbf {v} =\langle 2,4,-4\rangle

6{\sqrt {41}}

6{\sqrt {41}}

84. Find all vectors that satisfy the equation

\langle 1,1,1\rangle \times \mathbf {u} =\langle 0,1,1\rangle

None

85. Find the volume of the parallelepiped with edges given by position vectors

\langle 5,0,0\rangle

,

\langle 1,4,0\rangle

, and

\langle 2,2,7\rangle

140

140

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

\mathbf {F} =\langle 1,2,3\rangle

is applied to the wrench n units away from the origin.

\mathbf {\tau } =\langle 0,-3n,2n\rangle

, so the torque is directed along

\pm \langle 0,-3,2\rangle

\mathbf {\tau } =\langle 0,-3n,2n\rangle

, so the torque is directed along

\pm \langle 0,-3,2\rangle

Prove the following identities or show them false by giving a counterexample.

87.

\mathbf {u} \times (\mathbf {u} \times \mathbf {v} )=\mathbf {0}

False:

\mathbf {i} \times (\mathbf {i} \times \mathbf {j} )=-\mathbf {j}

False:

\mathbf {i} \times (\mathbf {i} \times \mathbf {j} )=-\mathbf {j}

88.

\mathbf {u} \cdot (\mathbf {v} \times \mathbf {w} )=\mathbf {w} \cdot (\mathbf {u} \times \mathbf {v} )

Once expressed in component form, both sides evaluate to

u_{1}v_{2}w_{3}-u_{1}v_{3}w_{2}+u_{2}v_{3}w_{1}-u_{2}v_{1}w_{3}+u_{3}v_{1}w_{2}-u_{3}v_{2}w_{1}

Once expressed in component form, both sides evaluate to

u_{1}v_{2}w_{3}-u_{1}v_{3}w_{2}+u_{2}v_{3}w_{1}-u_{2}v_{1}w_{3}+u_{3}v_{1}w_{2}-u_{3}v_{2}w_{1}

89.

(\mathbf {u} -\mathbf {v} )\times (\mathbf {u} +\mathbf {v} )=2(\mathbf {u} \times \mathbf {v} )

Failed to parse (unknown function "\begin{eqnarray}"): {\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}}

Calculus of Vector-Valued Functions

100. Differentiate

\mathbf {r} (t)=\langle te^{-t},t\ln t,t\cos(t)\rangle

.

\langle e^{-t}-te^{-t},\ln(t)+1,cos(t)-t\sin(t)\rangle

\langle e^{-t}-te^{-t},\ln(t)+1,cos(t)-t\sin(t)\rangle

101. Find a tangent vector for the curve

\mathbf {r} (t)=\langle 2t^{4},6t^{3/2},10/t\rangle

at the point

t=1

.

\langle 8,9,-10\rangle

\langle 8,9,-10\rangle

102. Find the unit tangent vector for the curve

\mathbf {r} (t)=\langle t,2,2/t\rangle ,\ t\neq 0

.

\displaystyle {\frac {\langle t^{2},0,-2\rangle }{\sqrt {t^{4}+4}}}

\displaystyle {\frac {\langle t^{2},0,-2\rangle }{\sqrt {t^{4}+4}}}

103. Find the unit tangent vector for the curve

\mathbf {r} (t)=\langle \sin(t),\cos(t),e^{-t}\rangle ,\ t\in [0,\pi ]

at the point

t=0

.

\displaystyle {\frac {\langle 1,0,-1\rangle }{\sqrt {2}}}

\displaystyle {\frac {\langle 1,0,-1\rangle }{\sqrt {2}}}

104. Find

\mathbf {r}

if

\mathbf {r} '(t)=\langle {\sqrt {t}},\cos(\pi t),4/t\rangle

and

\mathbf {r} (1)=\langle 2,3,4\rangle

.

\displaystyle \left\langle {\frac {2t^{3/2}+4}{3}},{\frac {\sin(\pi t)}{\pi }}+3,4\ln |t|+4\right\rangle

\displaystyle \left\langle {\frac {2t^{3/2}+4}{3}},{\frac {\sin(\pi t)}{\pi }}+3,4\ln |t|+4\right\rangle

105. Evaluate

\displaystyle \int _{0}^{\ln 2}(e^{-t}\mathbf {i} +2e^{2t}\mathbf {j} -4e^{t}\mathbf {k} )dt

\langle 1/2,3,-4\rangle

\langle 1/2,3,-4\rangle

Motion in Space

120. Find velocity, speed, and acceleration of an object if the position is given by

\mathbf {r} (t)=\langle 3\sin(t),5\cos(t),4\sin(t)\rangle

.

\mathbf {v} =\langle 3\cos(t),-5\sin(t),4\cos(t)\rangle

,

|\mathbf {v} |=5

,

\mathbf {a} =\langle -3\sin(t),-5\cos(t),-4\sin(t)\rangle

\mathbf {v} =\langle 3\cos(t),-5\sin(t),4\cos(t)\rangle

,

|\mathbf {v} |=5

,

\mathbf {a} =\langle -3\sin(t),-5\cos(t),-4\sin(t)\rangle

121. Find the velocity and the position vectors for

t\geq 0

if the acceleration is given by

\mathbf {a} (t)=\langle e^{-t},1\rangle ,\ \mathbf {v} (0)=\langle 1,0\rangle ,\ \mathbf {r} (0)=\langle 0,0\rangle

.

\mathbf {v} (t)=\langle 2-e^{-t},t\rangle

,

\mathbf {r} (t)=\langle e^{-t}+2t-1,t^{2}/2\rangle

\mathbf {v} (t)=\langle 2-e^{-t},t\rangle

,

\mathbf {r} (t)=\langle e^{-t}+2t-1,t^{2}/2\rangle

Length of Curves

Find the length of the following curves.

140.

\mathbf {r} (t)=\langle 4\cos(3t),4\sin(3t)\rangle ,\ t\in [0,2\pi /3].

8\pi

8\pi

141.

\mathbf {r} (t)=\langle 2+3t,1-4t,3t-4\rangle ,\ t\in [1,6].

5{\sqrt {34}}

5{\sqrt {34}}

Parametrization and Normal Vectors

142. Find a description of the curve that uses arc length as a parameter:

\mathbf {r} (t)=\langle t^{2},2t^{2},4t^{2}\rangle \ t\in [1,4].

\displaystyle \mathbf {r} (s)=\left({\frac {s}{\sqrt {21}}}+1\right)\langle 1,2,4\rangle

\displaystyle \mathbf {r} (s)=\left({\frac {s}{\sqrt {21}}}+1\right)\langle 1,2,4\rangle

143. Find the unit tangent vector T and the principal unit normal vector N for the curve

\mathbf {r} (t)=\langle t^{2},t\rangle .

Check that T⋅N=0.

\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}}}

\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}}}

Equations of Lines And Planes

160. Find an equation of a plane passing through points

(1,1,2),\ (1,2,2),\ (-1,0,1).

x-2z+3=0

x-2z+3=0

161. Find an equation of a plane parallel to the plane 2x−y+z=1 passing through the point (0,2,-2)

2x-y+z+4=0

2x-y+z+4=0

162. Find an equation of the line perpendicular to the plane x+y+2z=4 passing through the point (5,5,5).

\mathbf {r} (t)=\langle 5+t,5+t,5+2t\rangle

\mathbf {r} (t)=\langle 5+t,5+t,5+2t\rangle

163. Find an equation of the line where planes x+2y−z=1 and x+y+z=1 intersect.

\mathbf {r} (t)=\langle 1-3t,2t,t\rangle

\mathbf {r} (t)=\langle 1-3t,2t,t\rangle

164. Find the angle between the planes x+2y−z=1 and x+y+z=1.

\cos ^{-1}{\frac {2}{\sqrt {18}}}

\cos ^{-1}{\frac {2}{\sqrt {18}}}

165. Find the distance from the point (3,4,5) to the plane x+y+z=1.

{\frac {11}{3}}{\sqrt {3}}

{\frac {11}{3}}{\sqrt {3}}

Limits And Continuity

Evaluate the following limits.

180.

\displaystyle \lim _{(x,y)\rightarrow (1,-2)}{\frac {y^{2}+2xy}{y+2x}}

−2

181.

\displaystyle \lim _{(x,y)\rightarrow (4,5)}{\frac {{\sqrt {x+y}}-3}{x+y-9}}

1/6

At what points is the function f continuous?

182.

f(x,y)=\ln |x-y|

\{(x,y)\mid x\neq y\}

\{(x,y)\mid x\neq y\}

183.

f(x,y)=\displaystyle {\frac {\ln(x^{2}+y^{2})}{x-y+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 \lim _{(x,y)\rightarrow (0,0)}{\frac {4xy}{3x^{2}+y^{2}}}

The limit is 1 along the line y=x, and −1 along the line y=−x

185.

\displaystyle \lim _{(x,y)\rightarrow (0,0)}{\frac {y}{\sqrt {x^{2}-y^{2}}}}

The limit is 0 along the line y=0, and

1/{\sqrt {3}}

along the line x=2y

The limit is 0 along the line y=0, and

1/{\sqrt {3}}

along the line x=2y

186.

\displaystyle \lim _{(x,y)\rightarrow (0,0)}{\frac {x^{3}-y^{2}}{x^{3}+y^{2}}}

The limit is 1 along the line y=0, and −1 along the line x=0

187.

\displaystyle \lim _{(x,y)\rightarrow (0,0)}{\frac {x^{2}y^{2}+y^{6}}{x^{3}}}

The limit is 0 along any line of the form y=mx, and 2 along the parabola

x=y^{2}

The limit is 0 along any line of the form y=mx, and 2 along the parabola

x=y^{2}

Partial Derivatives

200. Find

\partial z/\partial x

if

\displaystyle z(x,y)={\frac {1}{\ln(xy)}}

{\frac {-1}{x(\ln(xy))^{2}}}

{\frac {-1}{x(\ln(xy))^{2}}}

201. Find all three partial derivatives of the function

\displaystyle f(x,y,z)=xe^{y^{2}+z}

\displaystyle f_{x}=e^{y^{2}+z},\ f_{y}=2xye^{y^{2}+z},\ f_{z}=f.

\displaystyle f_{x}=e^{y^{2}+z},\ f_{y}=2xye^{y^{2}+z},\ f_{z}=f.

Find the four second partial derivatives of the following functions.

202.

f(x,y)=\cos(xy)

f_{xx}=-y^{2}\cos(xy),\ f_{yy}=-x^{2}\cos(xy),\ f_{xy}=f_{yx}=-\sin(xy)-xy\cos(xy).

f_{xx}=-y^{2}\cos(xy),\ f_{yy}=-x^{2}\cos(xy),\ f_{xy}=f_{yx}=-\sin(xy)-xy\cos(xy).

203.

f(x,y)=xe^{y}

f_{xx}=0,\ f_{yy}=xe^{y},\ f_{xy}=f_{yx}=e^{y}.

f_{xx}=0,\ f_{yy}=xe^{y},\ f_{xy}=f_{yx}=e^{y}.

Chain Rule

Find $df/dt.$

220.

f(x,y)=x^{2}y-xy^{3},\ x(t)=t^{2},\ y(t)=t^{-2}

2t+4t^{-5}

2t+4t^{-5}

221.

f(x,y)={\sqrt {x^{2}+y^{2}}},\ x(t)=\cos(2t),\ y(t)=\sin(2t)

0

222.

\displaystyle f(x,y,z)={\frac {x-y}{y+z}},\ x(t)=t,\ \displaystyle y(t)=2t,\ z(t)=3t

0

Find $f_{s},\ f_{t}.$

223.

f(x,y)=\sin(x)\cos(2y),\ x=s+t,\ y=s-t

Failed to parse (syntax error): {\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 f(x,y,z)={\frac {x-z}{y+z}},\ x(t)=s+t,\ y(t)=st,\ z(t)=s-t

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 \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

V={\frac {1}{3}}x^{2}h

, where x is the side of the square base and h is the height of the pyramid. Suppose that

\displaystyle x(t)={\frac {t}{t+1}}

and

\displaystyle h(t)={\frac {1}{t+1}}

for

t\geq 0.

Find

V'(t).

\displaystyle {\frac {2t-t^{2}}{3(t+1)^{4}}}

\displaystyle {\frac {2t-t^{2}}{3(t+1)^{4}}}

Tangent Planes

Find an equation of a plane tangent to the given surface at the given point(s).

240.

xy\sin(z)=1,\ (1,2,\pi /6),\ (-1,-2,5\pi /6).

(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

(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

241.

z=x^{2}e^{x-y},\ (2,2,4),\ (-1,-1,1).

-8(x-2)+4(y-2)+z-4=0,\ x+y+z+1=0

-8(x-2)+4(y-2)+z-4=0,\ x+y+z+1=0

242.

z=\tan ^{-1}(x+y),\ (0,0,0).

x+y-z=0

x+y-z=0

243.

\sin(xyz)=1/2,\ (\pi ,1,1/6).

x+\pi y+6\pi z=3\pi

x+\pi y+6\pi z=3\pi

Maximum And Minimum Problems

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.

f(x,y)=x^{4}+2y^{2}-4xy

Local minima at (1,1) and (−1,−1), saddle at (0,0)

261.

f(x,y)=\tan ^{-1}(xy)

Saddle at (0,0)

262.

f(x,y)=2xye^{-x^{2}-y^{2}}

Saddle at (0,0), local maxima at

(\pm 1/{\sqrt {2}},\pm 1/{\sqrt {2}}),

local minima at

(\pm 1/{\sqrt {2}},\mp 1/{\sqrt {2}})

Saddle at (0,0), local maxima at

(\pm 1/{\sqrt {2}},\pm 1/{\sqrt {2}}),

local minima at

(\pm 1/{\sqrt {2}},\mp 1/{\sqrt {2}})

Find absolute maximum and minimum values of the function f on the set R.

263.

f(x,y)=x^{2}+y^{2}-2y+1,\ R=\{(x,y)\mid x^{2}+y^{2}\leq 4\}

Maximum of 9 at (0,−2) and minimum of 0 at (0,1)

264.

f(x,y)=x^{2}+y^{2}-2x-2y,

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)

265. Find the point on the plane x−y+z=2 closest to the point (1,1,1).

(4/3,2/3,4/3)

(4/3,2/3,4/3)

266. Find the point on the surface

z=x^{2}+y^{2}+10

closest to the plane

x+2y-z=0.

(0.5,1,11.25)

(0.5,1,11.25)

Double Integrals over Rectangular Regions

Evaluate the given integral over the region R.

280.

\displaystyle \iint _{R}(x^{2}+xy)dA,\ R=\{(x,y)\mid x\in [1,2],\ y\in [-1,1]\}

14/3

14/3

281.

\displaystyle \iint _{R}(xy\sin(x^{2}))dA,\ R=\{(x,y)\mid x\in [0,{\sqrt {\pi /2}}],\ y\in [0,1]\}

1/4

1/4

282.

\displaystyle \iint _{R}{\frac {x}{(1+xy)^{2}}}dA,\ R=\{(x,y)\mid x\in [0,4],\ y\in [1,2]\}

\ln(5/3)

\ln(5/3)

Evaluate the given iterated integrals.

283.

\displaystyle \int _{0}^{2}\int _{0}^{1}x^{5}y^{2}e^{x^{3}y^{3}}dydx

(e^{8}-9)/9

(e^{8}-9)/9

284.

\displaystyle \int _{1}^{4}\int _{0}^{2}e^{y{\sqrt {x}}}dydx

e^{4}-e^{2}-2

e^{4}-e^{2}-2

Double Integrals over General Regions

Evaluate the following integrals.

300.

\displaystyle \iint _{R}xydA,

R is bounded by x=0, y=2x+1, and y=5−2x.

2

2

301.

\displaystyle \iint _{R}(x+y)dA,

R is in the first quadrant and bounded by x=0,

y=x^{2},

and

y=8-x^{2}.

152/3

152/3

Use double integrals to compute the volume of the given region.

302. The solid in the first octant bound by the coordinate planes and the surface

z=8-x^{2}-2y^{2}.

4\pi {\sqrt {2}}

4\pi {\sqrt {2}}

303. The solid beneath the cylinder

z=y^{2}

and above the region

R=\{(x,y)\mid y\in [0,1],\ x\in [y,1]\}.

1/12

1/12

304. The solid bounded by the paraboloids

z=x^{2}+y^{2}

and

z=50-x^{2}-y^{2}.

625\pi

625\pi

Double Integrals in Polar Coordinates

320. Evaluate

\displaystyle \iint _{R}2xydA

for

R=\{(r,\theta )\mid r\in [1,3],\ \theta \in [0,\pi /2]\}

20

20

321. Find the average value of the function

f(r,\theta )=1/r^{2}

over the region

\{(r,\theta )\mid r\in [2,4]\}.

\displaystyle {\frac {\ln 2}{6}}

\displaystyle {\frac {\ln 2}{6}}

322. Evaluate

\displaystyle \int _{0}^{3}\int _{0}^{\sqrt {9-x^{2}}}{\sqrt {x^{2}+y^{2}}}dydx.

9\pi /2

9\pi /2

323. Evaluate

\displaystyle \iint _{R}{\frac {x-y}{x^{2}+y^{2}+1}}dA

if R is the unit disk centered at the origin.

0

0

Triple Integrals

340. Evaluate

\displaystyle \int _{1}^{\ln 8}\int _{0}^{\ln 4}\int _{0}^{\ln 2}e^{-x-y-2z}dxdydz.

\displaystyle {\frac {3}{16}}\left(e^{-2}-8^{-2}\right)

\displaystyle {\frac {3}{16}}\left(e^{-2}-8^{-2}\right)

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.

8

8

342. Find the volume of the solid in the first octant bounded by the cylinder

z=\sin(y)

for

y\in [0,\pi ]

, and the planes y=x and x=0.

\pi

\pi

343. Evaluate

\displaystyle \int _{0}^{1}\int _{y}^{2-y}\int _{0}^{2-x-y}xydzdxdy.

2/15

2/15

344. Rewrite the integral

\displaystyle \int _{0}^{1}\int _{-2}^{2}\int _{0}^{\sqrt {4-y^{2}}}dzdydx

in the order dydzdx.

\displaystyle \int _{0}^{1}\int _{0}^{2}\int _{-{\sqrt {4-z^{2}}}}^{\sqrt {4-z^{2}}}dydzdx

\displaystyle \int _{0}^{1}\int _{0}^{2}\int _{-{\sqrt {4-z^{2}}}}^{\sqrt {4-z^{2}}}dydzdx

Cylindrical And Spherical Coordinates

360. Evaluate the integral in cylindrical coordinates:

\displaystyle \int _{0}^{3}\int _{0}^{\sqrt {9-x^{2}}}\int _{0}^{\sqrt {x^{2}+y^{2}}}{\frac {1}{\sqrt {x^{2}+y^{2}}}}dzdydx

9\pi /4

9\pi /4

361. Find the mass of the solid cylinder

D=\{(r,\theta ,z)\mid r\in [0,3],\ z\in [0,2]\}

given the density function

\delta (r,\theta ,z)=5e^{-r^{2}}

10\pi (1-e^{-9})

10\pi (1-e^{-9})

362. Use a triple integral to find the volume of the region bounded by the plane z=0 and the hyperboloid

z={\sqrt {17}}-{\sqrt {1+x^{2}+y^{2}}}

\displaystyle {\frac {2\pi (1+7{\sqrt {17}})}{3}}

\displaystyle {\frac {2\pi (1+7{\sqrt {17}})}{3}}

363. If D is a unit ball, use a triple integral in spherical coordinates to evaluate

\iiint _{D}(x^{2}+y^{2}+z^{2})^{5/2}dV

\pi /2

\pi /2

364. Find the mass of a solid cone

\{(\rho ,\phi ,\theta )\mid \phi \leq \pi /3,\ z\in [0,4]\}

if the density function is

\delta (\rho ,\phi ,\theta )=5-z

128\pi

128\pi

365. Find the volume of the region common to two cylinders:

x^{2}+z^{2}=1,\ y^{2}+z^{2}=1

16/3

16/3

Center of Mass and Centroid

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.

{\frac {\langle 3,5,7\rangle }{4}}

{\frac {\langle 3,5,7\rangle }{4}}

381. Find the center of mass for a piece of wire with the density

\rho (x)=1+\sin(x)

for

x\in [0,\pi ].

\pi /2

\pi /2

382. Find the center of mass for a piece of wire with the density

\rho (x)=2-x^{2}/16

for

x\in [0,4].

9/5

9/5

383. Find the centroid of the region in the first quadrant bounded by the coordinate axes and

x^{2}+y^{2}=16.

\left({\frac {16}{3\pi }},{\frac {16}{3\pi }}\right)

\left({\frac {16}{3\pi }},{\frac {16}{3\pi }}\right)

384. Find the centroid of the region in the first quadrant bounded by

y=\ln(x)

,

y=0

, and

x=e

.

((e^{2}+1)/4,e/2-1)

((e^{2}+1)/4,e/2-1)

385. Find the center of mass for the region

\{(x,y)\mid x\in [0,4],y\in [0,2]\}

, with the density

\rho (x,y)=1+x/2.

(7/3,1)

(7/3,1)

386. Find the center of mass for the triangular plate with vertices (0,0), (0,4), and (4,0), with density

\rho (x,y)=1+x+y.

(16/11,16/11)

(16/11,16/11)

Vector Fields

One can sketch two-dimensional vector fields by plotting vector values, flow curves, and/or equipotential curves.

401. Find and sketch the gradient field

\mathbf {F} =\nabla \phi

for the potential function

\phi (x,y)={\sqrt {x^{2}+y^{2}}}

.

\mathbf {F} =\left\langle {\frac {x}{\sqrt {x^{2}+y^{2}}}},{\frac {y}{\sqrt {x^{2}+y^{2}}}}\right\rangle

\mathbf {F} =\left\langle {\frac {x}{\sqrt {x^{2}+y^{2}}}},{\frac {y}{\sqrt {x^{2}+y^{2}}}}\right\rangle

402. Find and sketch the gradient field

\mathbf {F} =\nabla \phi

for the potential function

\phi (x,y)=\sin(x)\sin(y)

for

|x|\leq \pi

and

|y|\leq \pi

.

\nabla \phi (x,y)=\langle \cos(x)\sin(y),\sin(x)\cos(y)\rangle

\nabla \phi (x,y)=\langle \cos(x)\sin(y),\sin(x)\cos(y)\rangle

403. Find the gradient field

\mathbf {F} =\nabla \phi

for the potential function

\phi (x,y,z)=e^{-z}\sin(x+y)

\mathbf {F} =e^{-z}\left\langle \cos(x+y),\cos(x+y),-\sin(x+y)\right\rangle

\mathbf {F} =e^{-z}\left\langle \cos(x+y),\cos(x+y),-\sin(x+y)\right\rangle

Line Integrals

420. Evaluate

\int _{C}(x^{2}+y^{2})ds

if C is the line segment from (0,0) to (5,5)

{\frac {250{\sqrt {2}}}{3}}

{\frac {250{\sqrt {2}}}{3}}

421. Evaluate

\int _{C}(x^{2}+y^{2})ds

if C is the circle of radius 4 centered at the origin

128\pi

128\pi

422. Evaluate

\int _{C}(y-z)ds

if C is the helix

\mathbf {r} (t)=\langle 3\cos(t),3\sin(t),t\rangle ,\ t\in [0,2\pi ]

-2{\sqrt {10}}\pi ^{2}

-2{\sqrt {10}}\pi ^{2}

423. Evaluate

\int _{C}\mathbf {F} \cdot d\mathbf {r}

if

\mathbf {F} =\langle x,y\rangle

and C is the arc of the parabola

\mathbf {r} (t)=\langle 4t,t^{2}\rangle ,\ t\in [0,1]

17/2

17/2

424. Find the work required to move an object from (1,1,1) to (8,4,2) along a straight line in the force field

\displaystyle \mathbf {F} ={\frac {\langle x,y,z\rangle }{x^{2}+y^{2}+z^{2}}}

\ln(2{\sqrt {7}})

\ln(2{\sqrt {7}})

Conservative Vector Fields

Determine if the following vector fields are conservative on $\mathbb {R} ^{2}.$

440.

\langle -y,x+y\rangle

No

441.

\langle 2x^{3}+xy^{2},2y^{3}+x^{2}y\rangle

Yes

Determine if the following vector fields are conservative on their respective domains in $\mathbb {R} ^{3}.$ When possible, find the potential function.

442.

\langle y,x,1\rangle

\phi (x,y,z)=xy+z

\phi (x,y,z)=xy+z

443.

\langle x^{3},2y,-z^{3}\rangle

\phi (x,y,z)=(x^{4}+4y^{2}-z^{4})/4

\phi (x,y,z)=(x^{4}+4y^{2}-z^{4})/4

Green's Theorem

460. Evaluate the circulation of the field

\mathbf {F} =\langle 2xy,x^{2}-y^{2}\rangle

over the boundary of the region above y=0 and below y=x(2-x) in two different ways, and compare the answers.

0

0

461. Evaluate the circulation of the field

\mathbf {F} =\langle 0,x^{2}+y^{2}\rangle

over the unit circle centered at the origin in two different ways, and compare the answers.

0

0

462. Evaluate the flux of the field

\mathbf {F} =\langle y,-x\rangle

over the square with vertices (0,0), (1,0), (1,1), and (0,1) in two different ways, and compare the answers.

0

0

Divergence And Curl

480. Find the divergence of

\langle 2x,4y,-3z\rangle

3

3

481. Find the divergence of

\displaystyle {\frac {\langle x,y,z\rangle }{1+x^{2}+y^{2}}}

\displaystyle {\frac {x^{2}+y^{2}+3}{(1+x^{2}+y^{2})^{2}}}

\displaystyle {\frac {x^{2}+y^{2}+3}{(1+x^{2}+y^{2})^{2}}}

482. Find the curl of

\langle x^{2}-y^{2},xy,z\rangle

\langle 0,0,3y\rangle

\langle 0,0,3y\rangle

483. Find the curl of

\langle z^{2}\sin(y),xz^{2}\cos(y),2xz\sin(y)\rangle

\mathbf {0}

\mathbf {0}

484. Prove that the general rotation field

\mathbf {F} =\mathbf {a} \times \mathbf {r}

, where

\mathbf {a}

is a non-zero constant vector and

\mathbf {r} =\langle x,y,z\rangle

, has zero divergence, and the curl of

\mathbf {F}

is

2\mathbf {a}

.

If

\mathbf {a} =\langle a_{1},a_{2},a_{3}\rangle

, then

$\mathbf {F} =\mathbf {a} \times \mathbf {r} =\langle a_{2}z-a_{3}y,a_{3}x-a_{1}z,a_{1}y-a_{2}x\rangle =\langle f,g,h\rangle$ , and then

$\nabla \cdot \mathbf {F} =\mathbf {f} _{x}+\mathbf {g} _{y}+\mathbf {h} _{z}=0+0+0=0,$

\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} .

If

\mathbf {a} =\langle a_{1},a_{2},a_{3}\rangle

, then

$\mathbf {F} =\mathbf {a} \times \mathbf {r} =\langle a_{2}z-a_{3}y,a_{3}x-a_{1}z,a_{1}y-a_{2}x\rangle =\langle f,g,h\rangle$ , and then

$\nabla \cdot \mathbf {F} =\mathbf {f} _{x}+\mathbf {g} _{y}+\mathbf {h} _{z}=0+0+0=0,$

\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} .

Surface Integrals

500. Give a parametric description of the plane

2x-4y+3z=16.

\langle u,v,(16-2u+4v)/3\rangle ,\ u,v\in \mathbb {R}

\langle u,v,(16-2u+4v)/3\rangle ,\ u,v\in \mathbb {R}

501. Give a parametric description of the hyperboloid

z^{2}=1+x^{2}+y^{2}.

\langle {\sqrt {v^{2}-1}}\cos(u),{\sqrt {v^{2}-1}}\sin(u),v\rangle ,\ u\in [0,2\pi ],\ |v|\geq 1

\langle {\sqrt {v^{2}-1}}\cos(u),{\sqrt {v^{2}-1}}\sin(u),v\rangle ,\ u\in [0,2\pi ],\ |v|\geq 1

502. Integrate

f(x,y,z)=xy

over the portion of the plane z=2−x−y in the first octant.

2/{\sqrt {3}}

2/{\sqrt {3}}

503. Integrate

f(x,y,z)=x^{2}+y^{2}

over the paraboloid

z=x^{2}+y^{2},\ z\in [0,4].

{\frac {(391{\sqrt {17}}+1)\pi }{60}}

{\frac {(391{\sqrt {17}}+1)\pi }{60}}

504. Find the flux of the field

\mathbf {F} =\langle x,y,z\rangle

across the surface of the cone

z^{2}=x^{2}+y^{2},\ z\in [0,1],

with normal vectors pointing in the positive z direction.

0

0

505. Find the flux of the field

\mathbf {F} =\langle -y,z,1\rangle

across the surface

y=x^{2},\ z\in [0,4],\ x\in [0,1],

with normal vectors pointing in the positive y direction.

-10

-10

Stokes' Theorem

520. Use a surface integral to evaluate the circulation of the field

\mathbf {F} =\langle x^{2}-z^{2},y,2xz\rangle

on the boundary of the plane

z=4-x-y

in the first octant.

{\frac {-128}{3}}

{\frac {-128}{3}}

521. Use a surface integral to evaluate the circulation of the field

\mathbf {F} =\langle y^{2},-z^{2},x\rangle

on the circle

\mathbf {r} (t)=\langle 3\cos(t),4\cos(t),5\sin(t)\rangle .

15\pi

15\pi

522. Use a line integral to find

\iint _{S}(\nabla \times F)\cdot \mathbf {n} dS

where

\mathbf {F} =\langle x,y,z\rangle

,

S

is the upper half of the ellipsoid

{\frac {x^{2}}{4}}+{\frac {y^{2}}{9}}+z^{2}=1

, and

\mathbf {n}

points in the direction of the z-axis.

0

0

523. Use a line integral to find

\iint _{S}(\nabla \times F)\cdot \mathbf {n} dS

where

\mathbf {F} =\langle 2y,-z,x-y-z\rangle

,

S

is the part of the sphere

x^{2}+y^{2}+z^{2}=25

for

3\leq z\leq 5

, and

\mathbf {n}

points in the direction of the z-axis.

-32\pi

-32\pi

Divergence Theorem

Compute the net outward flux of the given field across the given surface.

540.

\mathbf {F} =\langle x,-2y,3z\rangle

,

S

is a sphere of radius

{\sqrt {6}}

centered at the origin.

16{\sqrt {6}}\pi

16{\sqrt {6}}\pi

541.

\mathbf {F} =\langle x,2y,z\rangle

,

S

is the boundary of the tetrahedron in the first octant bounded by

x+y+z=1

2/3

2/3

542.

\mathbf {F} =\langle y+z,x+z,x+y\rangle

,

S

is the boundary of the cube

\{(x,y,z)\mid |x|\leq 1,|y|\leq 1,|z|\leq 1\}

0

0

543.

\mathbf {F} =\langle x,y,z\rangle

,

S

is the surface of the region bounded by the paraboloid

z=4-x^{2}-y^{2}

and the xy-plane.

24\pi

24\pi

544.

\mathbf {F} =\langle z-x,x-y,2y-z\rangle

,

S

is the boundary of the region between the concentric spheres of radii 2 and 4, centered at the origin.

-224\pi

-224\pi

545.

\mathbf {F} =\langle x,2y,3z\rangle

,

S

is the boundary of the region between the cylinders

x^{2}+y^{2}=1

and

x^{2}+y^{2}=4

and cut off by planes

z=0

and

z=8

144\pi

144\pi