Wize University Calculus 1 Textbook > Applications of Integration

Volumes of Revolution by Cylindrical Shells

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Volumes of Revolution by Cylindrical Shells


Volumes of Solids by Cylindrical Shells
If your cross-sectional area is a cylinder, then the surface of the lateral part of that cylinder is 2πrh2\pi rh2πrh. The associated volume obtained by revolving the region about the y-axis is given by 

V=2π∫abxf(x)dx\boxed{V=2 \pi\int_{a}^{b}  x f(x)  dx}V=2π∫ab​xf(x)dx​

More generally, think of the formula as
V=2π∫ab[radius][height]dx\boxed{V=2\pi\int_a^b[\text{radius}][\text{height}]dx}V=2π∫ab​[radius][height]dx​
This formula is useful when revolving around other lines parallel to the y-axis.

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Volumes Between Curves by Cylindrical Shells
If f(x)≥g(x)≥0f(x) \ge g(x) \ge 0f(x)≥g(x)≥0 then the volume of the solid of revolution obtained by  revolving the region between f(x)f(x)f(x) and g(x)g(x)g(x) betweenx=ax=ax=a and x=bx=bx=b around the y-axis is given by 

V=2π∫abx[f(x)−g(x)]dx\boxed{V=2\pi\int_a^bx[f(x)-g(x)]dx}V=2π∫ab​x[f(x)−g(x)]dx​

Wize Tip
This method can also be used for axes parallel to the x-axis (useful for when there is a "left" and "right" function, but no "top" and "bottom").

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Volumes about the X-Axis by Cylindrical Shells
If we'd like to use the Cylindrical Shell method about the x-axis, we need to write our equation as a function of yyy.

V=2π∫abyg(y)dy\boxed{V=2 \pi\int_{a}^{b}  yg(y)  dy}V=2π∫ab​yg(y)dy​

V=2π∫aby[f(y)−g(y)]dy\boxed{V=2\pi\int_a^by[f(y)-g(y)]dy}V=2π∫ab​y[f(y)−g(y)]dy​
These formulas is useful when revolving around other lines parallel to the y-axis.

Wize Concept
For the method of Cylindrical Shells the variables in the integral should be opposite of the axis of rotation. (revolving about the y-axis →use dx\rightarrow \text{use } dx→use dx, revolving about the x-axis →use dy\rightarrow \text{use } dy→use dy)

Practice: Volumes of Revolution
Write an integral representing the volume of the solid that is produced by revolving the region bounded between y=(x−1)4y=(x-1)^4y=(x−1)4 and 8x−y=88x-y=88x−y=8 about the line x=10x=10x=10. Do not compute the integral.

V=2π∫13(10−x)((8x−8)+(x−1)4)dx\displaystyle V = 2\pi\int^3_1(10 − x)((8x − 8) + (x − 1)^4) dxV=2π∫13​(10−x)((8x−8)+(x−1)4)dx

V=2π∫02(10−x)((8x−8)+(x−1)4)dx\displaystyle V = 2\pi\int^2_0(10 − x)((8x − 8) + (x − 1)^4) dxV=2π∫02​(10−x)((8x−8)+(x−1)4)dx

V=2π∫13(10−x)((8x−8)−(x−1)4)dx\displaystyle V = 2\pi\int^3_1(10 − x)((8x − 8) − (x − 1)^4) dxV=2π∫13​(10−x)((8x−8)−(x−1)4)dx

V=2π∫02(10−x)((8x−8)−(x−1)4)dx\displaystyle V = 2\pi\int^2_0(10 − x)((8x − 8) − (x − 1)^4) dxV=2π∫02​(10−x)((8x−8)−(x−1)4)dx

None of the above

I don't know

Practice: Volumes of Revolution
Write the integrals representing the volume of the solid that is produced by revolving the region bounded between y=exy=e^xy=ex and y=(e−1)x+1y=(e-1)x+1y=(e−1)x+1 about the line y=−1y=-1y=−1 using both the washer method and the method of cylindrical shells. Do not compute the integrals.

Washer: V=π∫01[(e−1)x+2]2−[ex+1]2dxV=\pi\int_0^1[(e−1)x+2]^2−[e^x+1]^2dxV=π∫01​[(e−1)x+2]2−[ex+1]2dx

Shells: V=2π∫1e2(y+1)[ln⁡y−(y−1)/(e−1)]dyV=2\pi\int_1^{e^2}(y+1)[\ln y−(y−1)/(e−1)]dyV=2π∫1e2​(y+1)[lny−(y−1)/(e−1)]dy

Washer: V=π∫02[(e−1)x+2]2−[ex+1]2dxV=\pi\int_0^2[(e−1)x+2]^2−[e^x+1]^2dxV=π∫02​[(e−1)x+2]2−[ex+1]2dx

Shells: V=2π∫1e2(y+1)[ln⁡y−(y−1)/(e+1)]dyV=2\pi\int_1^{e^2}(y+1)[\ln y−(y−1)/(e+1)]dyV=2π∫1e2​(y+1)[lny−(y−1)/(e+1)]dy

Washer: V=π∫02[(e−1)x+2]2−[ex+1]2dxV=\pi\int_0^2[(e−1)x+2]^2−[e^x+1]^2dxV=π∫02​[(e−1)x+2]2−[ex+1]2dx

Shells: V=2π∫1e(y+1)[ln⁡y−(y−1)/(e+1)]dyV=2\pi\int_1^e(y+1)[\ln y−(y−1)/(e+1)]dyV=2π∫1e​(y+1)[lny−(y−1)/(e+1)]dy

Washer: V=π∫01[(e−1)x+2]2−[ex+1]2dxV=\pi\int_0^1[(e−1)x+2]^2−[e^x+1]^2dxV=π∫01​[(e−1)x+2]2−[ex+1]2dx

Shells: V=2π∫1e(y+1)[ln⁡y−(y−1)/(e−1)]dyV=2\pi\int_1^e(y+1)[\ln y−(y−1)/(e−1)]dyV=2π∫1e​(y+1)[lny−(y−1)/(e−1)]dy

None of the above

I don't know

Extra Practice

Volumes of Revolution by Cylindrical Shells

Find the volume of the solid that is produced by revolving the region bounded between x=1+y2x=1+y^2x=1+y2, x=0x=0x=0, y=1y=1y=1, and y=2y=2y=2 about the xxx-axis.

Write the integrals representing the volume of the solid that is produced by revolving the region bounded between y=tan⁡3xy=\tan^3xy=tan3x, y=1y=1y=1, and x=0x=0x=0 about the line y=2y=2y=2 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−1)4y=(x-1)^4y=(x−1)4 and 8x−y=88x-y=88x−y=8 about the line x=10x=10x=10. Do not compute the integral.

Volumes of Revolution by Cylindrical Shells

Find the volume of the solid that is produced by revolving the region bounded between x=1+y2x=1+y^2x=1+y2, x=0x=0x=0, y=1y=1y=1, and y=2y=2y=2 about the xxx-axis.

Volumes of Revolution by Cylindrical Shells

Find the volume of the solid that is produced by revolving the region bounded between x=1+y2x=1+y^2x=1+y2, x=0x=0x=0, y=1y=1y=1, and y=2y=2y=2 about the xxx-axis.

Write the integrals representing the volume of the solid that is produced by revolving the region bounded between y=tan⁡3xy=\tan^3xy=tan3x, y=1y=1y=1, and x=0x=0x=0 about the line y=2y=2y=2 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−1)4y=(x-1)^4y=(x−1)4 and 8x−y=88x-y=88x−y=8 about the line x=10x=10x=10. Do not compute the integral.

Find the volume of revolution obtained by revolving around the yyy -axis the region of the first quadrant bounded by the x-axis and the curve y=3x2−x3y = 3x^2 -x^3y=3x2−x3. Use the method of cylindrical shells.

Using the method of shells, write an integral representing the volume of the solid that is produced by revolving the region bounded between y=(x−1)4y=(x-1)^4y=(x−1)4 and 8x−y=88x-y=88x−y=8 about the line x=10x=10x=10. 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⁡3xy=\tan^3xy=tan3x, y=1y=1y=1, and x=0x=0x=0 about the line y=2y=2y=2 using both the washer method and the method of cylindrical shells. Do not compute the integrals.

Write the integrals representing the volume of the solid that is produced by revolving the region bounded between y=exy=e^xy=ex and y=(e−1)x+1y=(e-1)x+1y=(e−1)x+1 about the line y=−1y=-1y=−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−1)4y=(x-1)^4y=(x−1)4 and 8x−y=88x-y=88x−y=8 about the line x=10x=10x=10. 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⁡3xy=\tan^3xy=tan3x, y=1y=1y=1, and x=0x=0x=0 about the line y=2y=2y=2 using both the washer method and the method of cylindrical shells. Do not compute the integrals.

Volumes of Revolution by Cylindrical Shells

Find the volume of the solid that is produced by revolving the region bounded between x=1+y2x=1+y^2x=1+y2, x=0x=0x=0, y=1y=1y=1, and y=2y=2y=2 about the xxx-axis.

Find the volume of revolution obtained by revolving around the y-axis the region of the first quadrant bounded by the x-axis and the curve y=3x2−x3y = 3x^2 -x^3y=3x2−x3. Use the method of cylindrical shells.

Find the volume of the solid that is produced by revolving the region bounded between y=x2/4y=x^2/4y=x2/4, y=0y=0y=0, and x=2x=2x=2 about the yyy-axis. Use the method of cylindrical shells.

Wize University Calculus 1 Textbook > Applications of Integration

Volumes of Revolution by Cylindrical Shells

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Volumes of Revolution by Cylindrical Shells

Volumes of Solids by Cylindrical Shells

Volumes Between Curves by Cylindrical Shells

Volumes about the X-Axis by Cylindrical Shells

Practice: Volumes of Revolution

Practice: Volumes of Revolution

Extra Practice

Volumes of Revolution by Cylindrical Shells

Volumes of Revolution by Cylindrical Shells

Volumes of Revolution by Cylindrical Shells

Volumes of Revolution by Cylindrical Shells