From Wikibooks, open books for an open world
22. Find
∫
0
∞
sin
(
x
)
cos
(
x
)
d
x
{\displaystyle \displaystyle \int _{0}^{\infty }\sin(x)\cos(x)dx}
40. Find the distance traveled by the particle with position function
p
(
t
)
=
(
2
sin
(
t
)
,
5
t
,
2
cos
(
t
)
)
{\displaystyle \displaystyle \mathbf {p} (t)=(2\sin(t),5t,2\cos(t))}
for
t
∈
[
−
10
,
10
]
{\displaystyle t\in [-10,10]}
.
20
29
{\displaystyle 20{\sqrt {29}}}
20
29
{\displaystyle 20{\sqrt {29}}}
41. Find the distance traveled by the particle with position function
p
(
t
)
=
(
12
t
,
8
t
3
/
2
,
3
t
2
)
{\displaystyle \displaystyle \mathbf {p} (t)=(12t,8t^{3/2},3t^{2})}
for
t
∈
[
0
,
1
]
{\displaystyle t\in [0,1]}
.
15
{\displaystyle 15}
15
{\displaystyle 15}
42. Find the distance traveled by the particle with position function
p
(
t
)
=
(
t
2
,
−
2
t
,
ln
(
t
)
)
{\displaystyle \displaystyle \mathbf {p} (t)=(t^{2},-2t,\ln(t))}
for
t
∈
[
1
,
e
]
{\displaystyle t\in [1,e]}
.
e
2
{\displaystyle e^{2}}
e
2
{\displaystyle e^{2}}
43. Find the arc length of the graph of the function
y
(
x
)
=
a
x
+
b
{\displaystyle \displaystyle y(x)=ax+b}
, where
a
≠
0
{\displaystyle a\neq 0}
, for
x
∈
[
c
,
d
]
{\displaystyle x\in [c,d]}
.
(
d
−
c
)
1
+
a
2
{\displaystyle (d-c){\sqrt {1+a^{2}}}}
(
d
−
c
)
1
+
a
2
{\displaystyle (d-c){\sqrt {1+a^{2}}}}
50. Find the area swept out by a particle moving along the parametrized curve
x
=
t
{\displaystyle x=t}
,
y
=
t
3
{\displaystyle y=t^{3}}
, for
t
∈
[
−
1
,
1
]
{\displaystyle t\in [-1,1]}
. Plot the curve and shade the area swept out before setting up the integral.
1
2
{\displaystyle {\frac {1}{2}}}
1
2
{\displaystyle {\frac {1}{2}}}
51. Find the area swept out by a particle moving along the parametrized curve
x
=
t
3
{\displaystyle x=t^{3}}
,
y
=
t
2
{\displaystyle y=t^{2}}
, for
t
∈
[
−
2
,
2
]
{\displaystyle t\in [-2,2]}
. Plot the curve and shade the area swept out before setting up the integral.
32
5
{\displaystyle {\frac {32}{5}}}
32
5
{\displaystyle {\frac {32}{5}}}
52. Plot the polar curve
r
=
sin
(
2
θ
)
{\displaystyle r=\sin(2\theta )}
for
θ
∈
[
0
,
2
π
]
{\displaystyle \theta \in [0,2\pi ]}
, and find the area enclosed by it.
π
2
{\displaystyle {\frac {\pi }{2}}}
π
2
{\displaystyle {\frac {\pi }{2}}}
53. Plot the polar curve
r
=
1
+
cos
(
θ
)
{\displaystyle r=1+\cos(\theta )}
for
θ
∈
[
0
,
2
π
]
{\displaystyle \theta \in [0,2\pi ]}
, and find the area enclosed by it.
3
2
π
{\displaystyle {\frac {3}{2}}\pi }
3
2
π
{\displaystyle {\frac {3}{2}}\pi }
54. Plot the polar curve
r
=
cos
(
5
θ
)
{\displaystyle r=\cos(5\theta )}
for
θ
∈
[
0
,
π
]
{\displaystyle \theta \in [0,\pi ]}
, and find the area enclosed by it.
π
4
{\displaystyle {\frac {\pi }{4}}}
π
4
{\displaystyle {\frac {\pi }{4}}}
60. Let
S
{\displaystyle S}
be the region in the first quadrant above the
x
{\displaystyle x}
-axis and below the curve
y
=
x
n
{\displaystyle y=x^{n}}
,
x
∈
[
0
,
1
]
{\displaystyle x\in [0,1]}
,
n
{\displaystyle n}
a positive ingeter. Find the volume of the solid obtained by revolving
S
{\displaystyle S}
about the
x
{\displaystyle x}
-axis.
π
2
n
+
1
{\displaystyle \displaystyle {\frac {\pi }{2n+1}}}
π
2
n
+
1
{\displaystyle \displaystyle {\frac {\pi }{2n+1}}}
61. Let
S
{\displaystyle S}
be the region above the line
y
=
x
−
2
{\displaystyle y=x-2}
and below the line
y
=
2
−
x
{\displaystyle y=2-x}
,
x
∈
[
0
,
2
]
{\displaystyle x\in [0,2]}
. Find the volume of the solid obtained by revolving
S
{\displaystyle S}
about the
y
{\displaystyle y}
-axis.
16
π
3
{\displaystyle \displaystyle {\frac {16\pi }{3}}}
16
π
3
{\displaystyle \displaystyle {\frac {16\pi }{3}}}
71. Find the mass of the thin plate lying in the
x
y
{\displaystyle xy}
-plane below the curve
y
=
1
−
x
2
{\displaystyle y=1-x^{2}}
and above the curve
y
=
x
2
−
1
{\displaystyle y=x^{2}-1}
if the area density is
δ
a
r
(
x
)
=
x
+
2
{\displaystyle \delta _{ar}(x)=x+2}
.
16
3
{\displaystyle {\frac {16}{3}}}
16
3
{\displaystyle {\frac {16}{3}}}
80. Find the center of mass of a thin wire extending from
x
=
1
{\displaystyle x=1}
to
x
=
e
{\displaystyle x=e}
along the
x -axis if the linear density of the wire is
δ
l
(
x
)
=
l
n
(
x
)
{\displaystyle \delta _{l}(x)=ln(x)}
.
e
2
+
1
4
{\displaystyle {\frac {e^{2}+1}{4}}}
e
2
+
1
4
{\displaystyle {\frac {e^{2}+1}{4}}}
81. Find the center of mass of a thin plate occupying the region
S
{\displaystyle S}
in the
xy -plane, if
S
{\displaystyle S}
is a region below the curve
y
=
x
2
{\displaystyle y=x^{2}}
and above the curve
y
=
x
3
{\displaystyle y=x^{3}}
, with
x
∈
[
0
,
1
]
{\displaystyle x\in [0,1]}
, and the area density of the plate is
δ
a
r
(
x
)
=
(
x
+
1
)
{\displaystyle \delta _{ar}(x)=(x+1)}
.
(
5
8
,
83
224
)
{\displaystyle \left({\frac {5}{8}},{\frac {83}{224}}\right)}
(
5
8
,
83
224
)
{\displaystyle \left({\frac {5}{8}},{\frac {83}{224}}\right)}
90. Suppose that we have a tank, which is a right circular cylinder of radius 1 meter and height 4 meters, and the tank is initially filled half-way. Find the amount of work required to pump all of the water to the top of the tank. Use
1000
k
g
/
m
3
{\displaystyle 1000\ \mathrm {kg} /\mathrm {m} ^{3}}
as the density of water and
g
=
9.8
m
/
s
2
{\displaystyle g=9.8\ \mathrm {m} /\mathrm {s} ^{2}}
as the gravity of Earth.
91. Suppose that a bucket is lifted to the top of a building 12 meters high at a constant rate of
1
m
/
s
{\displaystyle 1\ \mathrm {m} /\mathrm {s} }
. The initial weight of the bucket is
14
k
g
{\displaystyle 14\ \mathrm {kg} }
, and it is leaking sand at the rate of
0.2
k
g
/
s
{\displaystyle 0.2\ \mathrm {kg} /\mathrm {s} }
. Find the work required to lift the bucket. Use
g
=
9.8
m
/
s
2
{\displaystyle g=9.8\ \mathrm {m} /\mathrm {s} ^{2}}
as the gravity of Earth.
100. Find the Taylor polynomial of 8th degree, centered at zero, for the function
f
(
x
)
=
cos
(
x
)
{\displaystyle f(x)=\cos(x)}
, and make a guess about the corresponding Taylor series.
1
−
x
2
2
!
+
x
4
4
!
−
x
6
6
!
+
x
8
8
!
{\displaystyle 1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+{\frac {x^{8}}{8!}}}
1
−
x
2
2
!
+
x
4
4
!
−
x
6
6
!
+
x
8
8
!
{\displaystyle 1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+{\frac {x^{8}}{8!}}}
101. Find the Taylor polynomial of 8th degree, centered at zero, for the function
x
sin
(
x
)
{\displaystyle x\sin(x)}
, and make a guess about the corresponding Taylor series.
x
2
−
x
4
3
!
+
x
6
5
!
−
x
8
7
!
{\displaystyle x^{2}-{\frac {x^{4}}{3!}}+{\frac {x^{6}}{5!}}-{\frac {x^{8}}{7!}}}
x
2
−
x
4
3
!
+
x
6
5
!
−
x
8
7
!
{\displaystyle x^{2}-{\frac {x^{4}}{3!}}+{\frac {x^{6}}{5!}}-{\frac {x^{8}}{7!}}}
102. Find the Taylor series for the function
f
(
x
)
=
e
x
{\displaystyle f(x)=e^{x}}
centered at
a
=
1
{\displaystyle a=1}
.
∑
k
=
0
∞
e
(
x
−
1
)
k
k
!
{\displaystyle \sum _{k=0}^{\infty }{\frac {e(x-1)^{k}}{k!}}}
∑
k
=
0
∞
e
(
x
−
1
)
k
k
!
{\displaystyle \sum _{k=0}^{\infty }{\frac {e(x-1)^{k}}{k!}}}
103. Find the Taylor polynomial of 3rd degree for the function
f
(
x
)
=
x
{\displaystyle f(x)={\sqrt {x}}}
, centered at
a
=
1
{\displaystyle a=1}
.
1
+
(
x
−
1
)
2
−
(
x
−
1
)
2
8
+
(
x
−
1
)
3
16
{\displaystyle 1+{\frac {(x-1)}{2}}-{\frac {(x-1)^{2}}{8}}+{\frac {(x-1)^{3}}{16}}}
1
+
(
x
−
1
)
2
−
(
x
−
1
)
2
8
+
(
x
−
1
)
3
16
{\displaystyle 1+{\frac {(x-1)}{2}}-{\frac {(x-1)^{2}}{8}}+{\frac {(x-1)^{3}}{16}}}
110. Find the area of the triangle with vertices
(
1
,
2
,
3
)
{\displaystyle (1,2,3)}
,
(
−
1
,
−
1
,
2
)
{\displaystyle (-1,-1,2)}
, and
(
3
,
0
,
−
3
)
{\displaystyle (3,0,-3)}
.
138
{\displaystyle {\sqrt {138}}}
138
{\displaystyle {\sqrt {138}}}
111. Find a standard equation for the plane containing both the point
(
5
,
0
,
2
)
{\displaystyle (5,0,2)}
and the line
(
x
,
y
,
z
)
=
(
3
,
0
,
1
)
+
t
(
2
,
1
,
2
)
{\displaystyle (x,y,z)=(3,0,1)+t(2,1,2)}
.
x
+
2
y
−
2
z
−
1
=
0
{\displaystyle x+2y-2z-1=0}
x
+
2
y
−
2
z
−
1
=
0
{\displaystyle x+2y-2z-1=0}
120. Let the position of the particle be given by
p
(
t
)
=
(
2
sin
(
t
)
,
2
cos
(
t
)
)
{\displaystyle \mathbf {p} (t)=(2\sin(t),2\cos(t))}
. Find velocity, acceleration, and speed of the particle as functions of
t
{\displaystyle t}
. Sketch the path of the particle, and draw the velocity vector
v
(
π
/
4
)
{\displaystyle \mathbf {v} (\pi /4)}
and the acceleration vector
a
(
π
/
4
)
{\displaystyle \mathbf {a} (\pi /4)}
.
v
(
t
)
=
(
2
cos
(
t
)
,
−
2
sin
(
t
)
)
{\displaystyle \mathbf {v} (t)=(2\cos(t),-2\sin(t))}
|
v
(
t
)
|
=
2
{\displaystyle |\mathbf {v} (t)|=2}
a
(
t
)
=
(
−
2
sin
(
t
)
,
−
2
cos
(
t
)
)
{\displaystyle \mathbf {a} (t)=(-2\sin(t),-2\cos(t))}
v
(
t
)
=
(
2
cos
(
t
)
,
−
2
sin
(
t
)
)
{\displaystyle \mathbf {v} (t)=(2\cos(t),-2\sin(t))}
|
v
(
t
)
|
=
2
{\displaystyle |\mathbf {v} (t)|=2}
a
(
t
)
=
(
−
2
sin
(
t
)
,
−
2
cos
(
t
)
)
{\displaystyle \mathbf {a} (t)=(-2\sin(t),-2\cos(t))}
121. Find the distance traveled by a particle between times
t
0
=
0
{\displaystyle t_{0}=0}
and
t
1
=
1
{\displaystyle t_{1}=1}
if the position function of the particle is
p
(
t
)
=
(
e
t
,
2
t
,
e
−
t
)
{\displaystyle \mathbf {p} (t)=(e^{t},{\sqrt {2}}t,e^{-t})}
.
e
−
e
−
1
{\displaystyle e-e^{-1}}
e
−
e
−
1
{\displaystyle e-e^{-1}}
130. Find the total derivative
d
p
f
(
v
)
{\displaystyle d_{\mathbf {p} }f(\mathbf {v} )}
of
f
(
x
,
y
,
z
)
=
x
z
2
+
sin
(
y
)
{\displaystyle f(x,y,z)=xz^{2}+\sin(y)}
at the point
p
=
(
−
1
,
0
,
2
)
{\displaystyle \mathbf {p} =(-1,0,2)}
, if
v
=
(
2
,
−
1
,
−
3
)
{\displaystyle \mathbf {v} =(2,-1,-3)}
.
19
{\displaystyle 19}
19
{\displaystyle 19}
131. Find the value of the directional derivative of
f
(
x
,
y
,
z
,
w
)
=
x
y
2
−
z
w
{\displaystyle f(x,y,z,w)=xy^{2}-zw}
at
p
=
(
1
,
2
,
3
,
4
)
{\displaystyle \mathbf {p} =(1,2,3,4)}
in the direction of the vector
(
1
,
3
,
5
,
1
)
{\displaystyle (1,3,5,1)}
.
−
7
/
6
{\displaystyle -7/6}
−
7
/
6
{\displaystyle -7/6}
(a)
(
10
,
12
)
2
61
{\displaystyle \displaystyle {\frac {(10,12)}{2{\sqrt {61}}}}}
2
61
{\displaystyle 2{\sqrt {61}}}
±
(
12
,
−
10
)
2
61
{\displaystyle \displaystyle \pm {\frac {(12,-10)}{2{\sqrt {61}}}}}
(a)
(
10
,
12
)
2
61
{\displaystyle \displaystyle {\frac {(10,12)}{2{\sqrt {61}}}}}
2
61
{\displaystyle 2{\sqrt {61}}}
±
(
12
,
−
10
)
2
61
{\displaystyle \displaystyle \pm {\frac {(12,-10)}{2{\sqrt {61}}}}}
![{\displaystyle \displaystyle \left[\ln(x^{2}+1)\right]^{2}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5697e8be87d688204cbc2454f0b3524b6e864fa)
![{\displaystyle t\in [-10,10]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40ee5e9412d3241fadffb3d70f26e750976d8552)
![{\displaystyle t\in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31a5c18739ff04858eecc8fec2f53912c348e0e5)
![{\displaystyle t\in [1,e]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bd4e66f116b1e45e09c307c06a608054c002274)
![{\displaystyle x\in [c,d]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aef10ac7d23aac26f4cb71a2ad9745f574ca5af1)
![{\displaystyle x\in [0,5/9].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/028fd162168450d4dbe10b6f4e45efa49789c9f3)
![{\displaystyle t\in [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95e826ba1c1444ac0d0e95163db2f676e081b538)
![{\displaystyle t\in [-2,2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d380c9b008be90a6bb5c4a6fb8e751b2f286377a)
![{\displaystyle \theta \in [0,2\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e90716b30df198360bcc41f8f9b230b76bf26163)
![{\displaystyle \theta \in [0,\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c833964ea08aa30df8b6f56664461a5499b38144)
![{\displaystyle x\in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64a15936df283add394ab909aa7a5e24e7fb6bb2)
![{\displaystyle x\in [0,2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8021e1d08eb2f89e01bc8a1310705553ef6cde78)
