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Volumes of Revolution by Cross Sections
Related Topics
Wize University Calculus 1 Textbook > Applications of Integration
Volumes of Revolution by Cross Sections (Washers)
5 Activities
Find the volume of the solid generated by rotating the region bounded by
y
=
x
2
y=x^2
y
=
x
2
and
y
=
x
y=x
y
=
x
about the
y
y
y
-axis.
π
6
\frac{\pi}{6}
6
π
π
3
\frac{\pi}{3}
3
π
2
π
6
\frac{\sqrt{2}\pi}{6}
6
2
π
2
π
3
\frac{\sqrt{2}\pi}{3}
3
2
π
3
π
2
\frac{3\pi}{2}
2
3
π
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Check Submission
More Volumes of Revolution by Cross Sections (Washers) Questions:
Volumes of Revolution by Cross Sections
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
5
−
x
2
y=5-x^2
y
=
5
−
x
2
and
y
=
4
y=4
y
=
4
about the line 𝑦 = 2.
Volumes of Revolution by Cross Sections
Find the integrals that represents the volume of the solid obtained by rotating the region bounded by
y
=
2
x
y=\frac{2}{\sqrt{x}}
y
=
x
2
,
y
=
1
y=1
y
=
1
, and
x
=
1
x=1
x
=
1
a.) around the
x
x
x
-axis
b.) around the
y
y
y
-axis
Write the integrals representing the volume of the solid that is produced by revolving the region bounded between
y
=
tan
3
x
y=\tan^3x
y
=
tan
3
x
,
y
=
1
y=1
y
=
1
, and
x
=
0
x=0
x
=
0
about the line
y
=
2
y=2
y
=
2
using both the washer method and the method of cylindrical shells. Do not compute the integrals.
Find the volume of the solid that is produced by revolving the region bounded by the x-axis and
y
=
x
sin
x
cos
x
y=\sqrt{x\sin x\cos x}
y
=
x
sin
x
cos
x
about the x-axis, between 0 and
π
/
2
\pi/2
π
/2
.
Write an integral representing the volume of the solid that is produced by revolving the region bounded between
y
=
x
y=x
y
=
x
and
y
=
3
x
−
x
2
y=3x-x^2
y
=
3
x
−
x
2
about the line
y
=
−
1
y=-1
y
=
−
1
. Do not compute the integral.
Find the volume of the solid that is produced by revolving
y
=
x
sin
x
cos
x
y=\sqrt{x\sin x\cos x}
y
=
x
sin
x
cos
x
about the
x
x
x
-axis, between 0 and
π
/
2
\pi/2
π
/2
.
Find the volume of the solid that is produced by revolving
y
=
x
sin
x
cos
x
y=\sqrt{x\sin x\cos x}
y
=
x
sin
x
cos
x
about the
x
x
x
-axis, between 0 and
π
/
2
\pi/2
π
/2
.
Find the volume of the solid that is produced by revolving
y
=
x
sin
x
cos
x
y=\sqrt{x\sin x\cos x}
y
=
x
sin
x
cos
x
about the
x
x
x
-axis, between 0 and
π
/
2
\pi/2
π
/2
.
Let
R
R
R
be the region in the first quadrant bounded by the graph of
y
=
e
x
y = e^x
y
=
e
x
, the line
y
=
2
y = 2
y
=
2
and the
y
y
y
-axis; let
S
S
S
be the solid of revolution obtained by revolving
R
R
R
about the
x
x
x
-axis. Compute the volume of
S
S
S
with the disc/washer method.
Volumes of Revolution by Cross Sections
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
5
−
x
2
y=5-x^2
y
=
5
−
x
2
and
y
=
4
y=4
y
=
4
about the line 𝑦 = 2.
Volumes of Revolution by Cross Sections
Find the integrals that represents the volume of the solid obtained by rotating the region bounded by
y
=
2
x
y=\frac{2}{\sqrt{x}}
y
=
x
2
,
y
=
1
y=1
y
=
1
, and
x
=
1
x=1
x
=
1
a.) around the
x
x
x
-axis
b.) around the
y
y
y
-axis
Q
:
\bf{Q:}
Q
:
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
5
−
x
2
y=5-x^2
y
=
5
−
x
2
and
y
=
4
y=4
y
=
4
about the line
y
=
2
y=2
y
=
2
.
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
e
x
y=e^x
y
=
e
x
,
y
=
1
y=1
y
=
1
and
x
=
4
x=4
x
=
4
, about the
x
x
x
-axis and then about the
y
y
y
-axis. (For the first part, enter the answer. For the second part enter the integral only, do not solve again. Factor out the pi.)
Write the integral that represents the volume of the solid generated by revolving the region bounded by
y
=
x
3
y=\sqrt[3]{x}
y
=
3
x
and
y
=
x
4
y=\dfrac{x}{4}
y
=
4
x
that lies in the first quadrant, about the
y
y
y
-axis. Use the method of disks/washers. Do not solve.
Volumes of Revolution by Cross Sections
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
5
−
x
2
y=5-x^2
y
=
5
−
x
2
and
y
=
4
y=4
y
=
4
about the line 𝑦 = 2.
Volumes of Revolution by Cross Sections
Find the integrals that represents the volume of the solid obtained by rotating the region bounded by
y
=
2
x
y=\frac{2}{\sqrt{x}}
y
=
x
2
,
y
=
1
y=1
y
=
1
, and
x
=
1
x=1
x
=
1
a.) around the
x
x
x
-axis
b.) around the
y
y
y
-axis
Write an integral representing the volume of the solid that is produced by revolving the region bounded between
y
=
x
y=x
y
=
x
and
y
=
3
x
−
x
2
y=3x-x^2
y
=
3
x
−
x
2
about the line
y
=
−
1
y=-1
y
=
−
1
. Do not compute the integral.
Practice: Volume of Cone
Q:
\textbf{Q:}
Q:
Derive the formula for the volume of a right circular cone whose altitude is
h
h
h
and whose radius of the base is
r
r
r
.
Practice: Rotate about Horizontal Line
Q:
\textbf{Q:}
Q:
Find the volume of the solid generated when the region enclosed by the curves
y
=
x
2
y = x^2
y
=
x
2
and
y
=
x
+
2
y = x + 2
y
=
x
+
2
is revolved about the line
y
=
−
1.
y = −1.
y
=
−
1.
Practice: Rotate about y-axis
Q:
\textbf{Q:}
Q:
Consider the region
R
R
R
of the first quadrant bounded by the curves
y
2
=
x
y^2 = x
y
2
=
x
and
y
=
x
4
y = x^4
y
=
x
4
. Find the volume of the region created by revolving
R
R
R
around the y-axis. Use the method of cross-sections.
Practice: Rotate about x-axis
Q:
\textbf{Q:}
Q:
Find the volume of the solid obtained by revolving the region enclosed by
y
=
4
−
5
x
2
2
y=4-\dfrac{5x^2}{2}
y
=
4
−
2
5
x
2
and
y
=
x
2
2
+
1
y=\dfrac{x^2}{2}+1
y
=
2
x
2
+
1
around the
x
x
x
-axis.
Practice: Rotate about x-axis
Q:
\textbf{Q:}
Q:
Find the volume of the solid obtained by revolving the region between
y
=
x
2
+
1
y=x^2+1
y
=
x
2
+
1
and
y
=
3
−
x
y=3-x
y
=
3
−
x
around the
x
x
x
-axis.
Practice: Volumes of Revolution (~F2018 Final Q43)
Practice: Volumes of Revolution
Find the integrals that represents the volume of the solid obtained by rotating the region bounded by
y
=
2
x
y=\frac{2}{\sqrt{x}}
y
=
x
2
,
y
=
1
y=1
y
=
1
, and
x
=
1
x=1
x
=
1
a.) around the
x
x
x
-axis
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
5
−
x
2
y=5-x^2
y
=
5
−
x
2
and
y
=
4
y=4
y
=
4
about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
5
−
x
2
y=5-x^2
y
=
5
−
x
2
and
y
=
4
y=4
y
=
4
about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
5
−
x
2
y=5-x^2
y
=
5
−
x
2
and
y
=
4
y=4
y
=
4
about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
5
−
x
2
y=5-x^2
y
=
5
−
x
2
and
y
=
4
y=4
y
=
4
about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
5
−
x
2
y=5-x^2
y
=
5
−
x
2
and
y
=
4
y=4
y
=
4
about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
5
−
x
2
y=5-x^2
y
=
5
−
x
2
and
y
=
4
y=4
y
=
4
about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
5
−
x
2
y=5-x^2
y
=
5
−
x
2
and
y
=
4
y=4
y
=
4
about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
5
−
x
2
y=5-x^2
y
=
5
−
x
2
and
y
=
4
y=4
y
=
4
about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
5
−
x
2
y=5-x^2
y
=
5
−
x
2
and
y
=
4
y=4
y
=
4
about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
5
−
x
2
y=5-x^2
y
=
5
−
x
2
and
y
=
4
y=4
y
=
4
about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
5
−
x
2
y=5-x^2
y
=
5
−
x
2
and
y
=
4
y=4
y
=
4
about the line 𝑦 = 2.
volume around y
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
x
y=\sqrt{x}
y
=
x
and
x
=
3
y
x=3y
x
=
3
y
around the 𝑦-axis.
volume around y
Find the volume of the solid generated by rotating the region bounded by
y
=
x
2
y=x^2
y
=
x
2
and
y
=
x
y=x
y
=
x
about the
y
y
y
-axis.
(a)
π
3
\frac{\pi}{3}
3
π
(b)
π
12
\frac{\pi}{12}
12
π
(c)
π
2
\frac{\pi}{2}
2
π
(d)
3
π
4
\frac{3\pi}{4}
4
3
π
(e)
π
6
\frac{\pi}{6}
6
π
Write the integrals representing the volume of the solid that is produced by revolving the region bounded between
y
=
tan
3
x
y=\tan^3x
y
=
tan
3
x
,
y
=
1
y=1
y
=
1
, and
x
=
0
x=0
x
=
0
about the line
y
=
2
y=2
y
=
2
using both the washer method and the method of cylindrical shells. Do not compute the integrals.
Find the volume of the solid that is produced by revolving the region bounded by the x-axis and
y
=
x
sin
x
cos
x
y=\sqrt{x\sin x\cos x}
y
=
x
sin
x
cos
x
about the x-axis, between 0 and
π
/
2
\pi/2
π
/2
.
Write an integral representing the volume of the solid that is produced by revolving the region bounded between
y
=
x
y=x
y
=
x
and
y
=
3
x
−
x
2
y=3x-x^2
y
=
3
x
−
x
2
about the line
y
=
−
1
y=-1
y
=
−
1
. Do not compute the integral.
Volumes of Revolution by Cross Sections
Evaluate the volume obtained by rotating the region enclosed by the curves
y
=
e
x
,
y
=
1
,
x
=
e
y=e^x,\ \ y=1,\ \ x=e
y
=
e
x
,
y
=
1
,
x
=
e
around the x-axis
Write the integral that represents the volume of the solid generated by revolving the region bounded by
y
=
x
3
y=\sqrt[3]{x}
y
=
3
x
and
y
=
x
4
y=\dfrac{x}{4}
y
=
4
x
that lies in the first quadrant, about the
y
y
y
-axis. Use the method of disks/washers. Do not solve.
Write the integrals representing the volume of the solid that is produced by revolving the region bounded between
y
=
tan
3
x
y=\tan^3x
y
=
tan
3
x
,
y
=
1
y=1
y
=
1
, and
x
=
0
x=0
x
=
0
about the line
y
=
2
y=2
y
=
2
using both the washer method and the method of cylindrical shells. Do not compute the integrals.
Find the volume of the solid that is produced by revolving
y
=
x
sin
x
cos
x
y=\sqrt{x\sin x\cos x}
y
=
x
sin
x
cos
x
about the
x
x
x
-axis, between 0 and
π
/
2
\pi/2
π
/2
.
Write an integral representing the volume of the solid that is produced by revolving the region bounded between
y
=
x
y=x
y
=
x
and
y
=
3
x
−
x
2
y=3x-x^2
y
=
3
x
−
x
2
about the line
y
=
−
1
y=-1
y
=
−
1
. Do not compute the integral.
Write an integral representing the volume of the solid that is produced by revolving the region bounded between
y
=
x
y=x
y
=
x
and
y
=
3
x
−
x
2
y=3x-x^2
y
=
3
x
−
x
2
about the line
y
=
−
1
y=-1
y
=
−
1
. Do not compute the integral.
Find the volume of the solid obtained by rotating the region bounded by
x
=
5
/
y
x=5/y
x
=
5/
y
,
y
=
1
y=1
y
=
1
and
x
=
2
x=2
x
=
2
about the y-axis.
volumeint
Write the integral that represents the volume of the solid obtained by rotating the region bounded by
y
=
5
x
y=\frac{5}{x}
y
=
x
5
,
y
=
1
y=1
y
=
1
and
x
=
2
x=2
x
=
2
about the x-axis.
Write the integral that represents the volume of the solid generated by revolving the region bounded by
y
=
x
3
y=\sqrt[3]{x}
y
=
3
x
and
y
=
x
4
y=\dfrac{x}{4}
y
=
4
x
that lies in the first quadrant, about the
y
y
y
-axis. Use the method of disks/washers. Do not solve.
Q
:
\bf{Q:}
Q
:
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
5
−
x
2
y=5-x^2
y
=
5
−
x
2
and
y
=
4
y=4
y
=
4
about the line
y
=
2
y=2
y
=
2
.
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
e
x
y=e^x
y
=
e
x
,
y
=
1
y=1
y
=
1
and
x
=
4
x=4
x
=
4
, about the
x
x
x
-axis and then about the
y
y
y
-axis. (For the first part, enter the answer. For the second part enter the integral only, do not solve again. Factor out the pi.)
Practice: Rotate about x-axis
Q:
\textbf{Q:}
Q:
Find the volume of the solid obtained by revolving the region between
y
=
x
2
+
1
y=x^2+1
y
=
x
2
+
1
and
y
=
3
−
x
y=3-x
y
=
3
−
x
around the
x
x
x
-axis.
Practice: Rotate about Horizontal Line
Q:
\textbf{Q:}
Q:
Find the volume of the solid generated when the region enclosed by the curves
y
=
x
2
y = x^2
y
=
x
2
and
y
=
x
+
2
y = x + 2
y
=
x
+
2
is revolved about the line
y
=
−
1.
y = −1.
y
=
−
1.
Practice: Rotate about y-axis
Q:
\textbf{Q:}
Q:
Consider the region
R
R
R
of the first quadrant bounded by the curves
y
2
=
x
y^2 = x
y
2
=
x
and
y
=
x
4
y = x^4
y
=
x
4
. Find the volume of the region created by revolving
R
R
R
around the y-axis. Use the method of cross-sections.
Practice: Volume of Cone
Q:
\textbf{Q:}
Q:
Derive the formula for the volume of a right circular cone whose altitude is
h
h
h
and whose radius of the base is
r
r
r
.
Practice: Rotate about x-axis
Q:
\textbf{Q:}
Q:
Find the volume of the solid obtained by revolving the region enclosed by
y
=
4
−
5
x
2
2
y=4-\dfrac{5x^2}{2}
y
=
4
−
2
5
x
2
and
y
=
x
2
2
+
1
y=\dfrac{x^2}{2}+1
y
=
2
x
2
+
1
around the
x
x
x
-axis.
Write the integrals representing the volume of the solid that is produced by revolving the region bounded between
y
=
e
x
y=e^x
y
=
e
x
and
y
=
(
e
−
1
)
x
+
1
y=(e-1)x+1
y
=
(
e
−
1
)
x
+
1
about the line
y
=
−
1
y=-1
y
=
−
1
using both the washer method and the method of cylindrical shells. Do not compute the integrals.
Write an integral representing the volume of the solid that is produced by revolving the region bounded between
y
=
x
y=x
y
=
x
and
y
=
3
x
−
x
2
y=3x-x^2
y
=
3
x
−
x
2
about the line
y
=
−
1
y=-1
y
=
−
1
. Do not compute the integral.
Write the integrals representing the volume of the solid that is produced by revolving the region bounded between
y
=
tan
3
x
y=\tan^3x
y
=
tan
3
x
,
y
=
1
y=1
y
=
1
, and
x
=
0
x=0
x
=
0
about the line
y
=
2
y=2
y
=
2
using both the washer method and the method of cylindrical shells. Do not compute the integrals.
Compute the volume of the solid formed when the area between
f
(
x
)
=
x
2
+
1
f\left(x\right)=x^2+1
f
(
x
)
=
x
2
+
1
,
x
=
0
x=0
x
=
0
,
x
=
2
x=2
x
=
2
, and
y
=
0
y=0
y
=
0
is rotated around the
y
-axis.
Let
R
R
R
be the region in the first quadrant bounded by the graph of
y
=
e
x
y = e^x
y
=
e
x
, the line
y
=
2
y = 2
y
=
2
and the
y
y
y
-axis; let
S
S
S
be the solid of revolution obtained by revolving
R
R
R
about the
x
x
x
-axis. Compute the volume of
S
S
S
with the disc/washer method.
Write the integral that represents the volume of the solid obtained by rotating the region bounded by 𝑦 = ln 𝑥 , 𝑦 = 2 and 𝑥 = 10 around the 𝑥-axis. Do not solve.
Write the integral that represents the volume of the solid generated by revolving the region bounded by 𝑦 =
3
x
3\sqrt{x}
3
x
and 𝑦 =
x
4
\frac{x}{4}
4
x
that lies in the first quadrant (top-right corner) about the 𝑦-axis. Do not solve.
Find the volume of the solid that is produced by revolving
y
=
x
sin
x
cos
x
y=\sqrt{x\sin x\cos x}
y
=
x
sin
x
cos
x
about the
x
x
x
-axis, between 0 and
π
/
2
\pi/2
π
/2
.
Volumes of Revolution by Cross Sections
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
5
−
x
2
y=5-x^2
y
=
5
−
x
2
and
y
=
4
y=4
y
=
4
about the line 𝑦 = 2.
Volumes of Revolution by Cross Sections
Find the integrals that represents the volume of the solid obtained by rotating the region bounded by
y
=
2
x
y=\frac{2}{\sqrt{x}}
y
=
x
2
,
y
=
1
y=1
y
=
1
, and
x
=
1
x=1
x
=
1
a.) around the
x
x
x
-axis
b.) around the
y
y
y
-axis
volume around y
Find the volume of the solid obtained by rotating the region bounded by the curves
y
=
x
y=\sqrt{x}
y
=
x
and
x
=
3
y
x=3y
x
=
3
y
around the 𝑦-axis.
Volumes of Revolution by Cross Sections
Which of the following integrals represent the volume of the solid generated by revolving the area bounded by
y
=
x
+
2
y=x+2
y
=
x
+
2
,
y
=
x
2
y=x^2
y
=
x
2
and
x
=
0
x=0
x
=
0
around the y-axis.