Wize University Calculus 2 Textbook > Applications of Integrals

Volumes of Revolution (Disks/Washer Method)

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Volumes of Revolution (Disk/Washer Method)
The question will ask you to compute the volume of the solid obtained by rotating a bouonded region around the x or y axis (or a line parallel to one of these axes).

Steps for finding the volume
1. Draw the curve


2. Rearrange the curve equations
If we're rotating around the x-axis: set up the equations as y=...y=...y=...
If we're rotating around the y-axis: set up the equations as x=...x=...x=...
3. Find the integral bounds
These could be given by the question
Otherwise set the two curve equations equal one another and solve for the points of intersection
4. Set up the volume integral and solve
If we're rotating around the x-axis: V=∫x=ax=bπ[outer radius2−inner radius2]dx\displaystyle V=\int_{x=a}^{x=b}\pi\left[\text{outer radius}^2-\text{inner radius}^2\right]dxV=∫x=ax=b​π[outer radius2−inner radius2]dx
If we're rotating around the y-axis: V=∫y=cy=dπ[outer radius2−inner radius2]dy\displaystyle V=\int_{y=c}^{y=d}\pi\left[\text{outer radius}^2-\text{inner radius}^2\right]dyV=∫y=cy=d​π[outer radius2−inner radius2]dy

Rotating around a line
Rotating around y=ky=ky=k (parallel to the x-axis) or x=kx=kx=k  (parallel to the y-axis): follow the same steps but make an adjustment for your outer and inner radius

Practice: Volumes of Revolution
Find the integral that represents the volume of the solid obtained by revolving the region enclosed by y=4−5x22y=4-\frac{5x^2}{2}y=4−25x2​ and y=x22+1y=\frac{x^2}{2}+1y=2x2​+1 about the xxx-axis.

V=π∫−11[(3−3x2)2]dx\displaystyle V=\pi\int_{-1}^1\left[\left(3-3x^2\right)^2\right]dxV=π∫−11​[(3−3x2)2]dx

V=π∫−11[(4−5x22)2−(x22+1)2]dx\displaystyle V=\pi\int_{-1}^1\left[\left(4-\frac{5x^2}{2}\right)^2-\left(\frac{x^2}{2}+1\right)^2\right]dxV=π∫−11​[(4−25x2​)2−(2x2​+1)2]dx

V=π∫−11[(x22+1)2−(4−5x22)2]dx\displaystyle V=\pi\int_{-1}^1\left[\left(\frac{x^2}{2}+1\right)^2-\left(4-\frac{5x^2}{2}\right)^2\right]dxV=π∫−11​[(2x2​+1)2−(4−25x2​)2]dx

V=π∫−10(x22+1)2dx−π∫01(4−5x22)2dx\displaystyle V=\pi\int_{-1}^0\left(\frac{x^2}{2}+1\right)^2dx-\pi\int_0^1\left(4-\frac{5x^2}{2}\right)^2dxV=π∫−10​(2x2​+1)2dx−π∫01​(4−25x2​)2dx

V=π∫−10(4−5x22)2dx−π∫01(x22+1)2dx\displaystyle V=\pi\int_{-1}^0\left(4-\frac{5x^2}{2}\right)^2dx-\pi\int_0^1\left(\frac{x^2}{2}+1\right)^2dxV=π∫−10​(4−25x2​)2dx−π∫01​(2x2​+1)2dx

I don't know

Practice: Volumes of Revolution
Find the integrals that represents the volume of the solid obtained by rotating the region bounded by y=2xy=\frac{2}{\sqrt{x}}y=x​2​, y=1y=1y=1, and x=1x=1x=1 around the yyy-axis.

V=π∫01[4x−1]dx\displaystyle V=\pi\int_0^1\left[\frac{4}{ x}-1\right]dxV=π∫01​[x4​−1]dx

V=π∫02[1−4x]dx\displaystyle V=\pi\int_0^2\left[1-\frac{4}{ x}\right]dxV=π∫02​[1−x4​]dx

V=π∫01[16y4−1]dy\displaystyle V=\pi\int_0^1\left[\frac{16}{y^4}-1\right]dyV=π∫01​[y416​−1]dy

V=π∫12[1−16y4]dy\displaystyle V=\pi\int_1^2\left[1-\frac{16}{y^4}\right]dyV=π∫12​[1−y416​]dy

V=π∫12[16y4−1]dy\displaystyle V=\pi\int_1^2\left[\frac{16}{y^4}-1\right]dyV=π∫12​[y416​−1]dy

I don't know

Practice: Volumes of Revolution
Write the integral that represents the volume generated by rotating the region bounded by y=xy=\sqrt{x}y=x​ and y=x2y=x^2y=x2 around the line x=−1x=-1x=−1.

Extra Practice

Find the volume of the solid that is produced by revolving y=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x}y=xsinxcosx​ about the xxx-axis, between 0 and π/2\pi/2π/2.

Find the volume of the solid that is produced by revolving y=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x}y=xsinxcosx​ about the xxx-axis, between 0 and π/2\pi/2π/2.

Find the volume of the solid that is produced by revolving y=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x}y=xsinxcosx​ about the xxx-axis, between 0 and π/2\pi/2π/2.

Find the volume of the solid that is produced by revolving y=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x}y=xsinxcosx​ about the xxx-axis, between 0 and π/2\pi/2π/2.

Wize University Calculus 2 Textbook > Applications of Integrals

Volumes of Revolution (Disks/Washer Method)

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Volumes of Revolution (Disk/Washer Method)

Practice: Volumes of Revolution

Practice: Volumes of Revolution

Practice: Volumes of Revolution

Extra Practice