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Wize University Calculus 1 Textbook > Applications of Integration

Volumes of Revolution by Cross Sections (Washers)

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Volumes of Revolution by Cross Sections (Washers)

While we commonly use integrals to find area under curves. If we imagine the graph "coming out of the page" and revolving in 3-D, we can extend our concepts of calculating area to calculating Volumes of Revolution.

Volumes of Solids by Cross Sections



The volume VV of a solid between x=ax=a and x=bx=b having cross-sectional area A(x)A(x)is rotated about the x-axis is
V=∫abA(x)dx\boxed{V=\int_a^bA(x)dx}
The cross-section of the solid generated in the plane perpendicular to the x-axis.

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Volumes of Solids by Washers

If your cross-sectional area is a disk, then the area of that disk will be πr2\pi r^2. For a functionf(x)f(x)the cross-sectional area isA(x)=π[f(x)]2A(x)=\pi[f(x)]^2.


For a function f(x)f(x), the volume of the solid of revolution obtained by revolving the region betweenf(x)f(x)and the x-axis between x=ax=a and x=bx=b about the x-axis is given by
V=∫abπ[f(x)]2dx\boxed{V=\int^b_a\pi [f(x)]^2dx}
This formula is useful for revolving about lines parallel to the x-axis.
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Volumes Between Curves by Washers





If f(x)≥g(x)≥0f(x)\ge g(x)\ge0 then the volume of the solid of revolution obtained by revolving the region between f(x)f\left(x\right) and g(x)g\left(x\right) between x=ax=a and x=bx=b about the x-axis is given by
V=∫abπ([f(x)]2−[g(x)]2)dx\boxed{V=\int_a^b\pi([f(x)]^2-[g(x)]^2)dx}

More generally, think of the formula as
V=∫abπ([outer radius]2−[inner radius]2)dx\boxed{V=\int_a^b\pi([\text{outer\ radius}]^2-[\text{inner\ radius}]^2)dx}


Watch Out!
It may be necessary to compute the points of intersection in order to obtain aa and bb.


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Volumes about the Y-Axis using by Washers

If we'd like to use the Washer method about the y-axis, we need to write our equation as a function of yy.

V=∫abπ[g(y)]2dy\boxed{V=\int^b_a\pi [g(y)]^2dy}

V=∫abπ([f(y)]2−[g(y)]2)dy\boxed{V=\int_a^b\pi([f(y)]^2-[g(y)]^2)dy}
These formulas is useful for revolving about lines parallel to the y-axis.

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

Watch Out!
Be careful to consider the inner and outer radii; this depends on the axis of rotation.

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Example: Volumes of Revolution

Find the volume of the solid obtained by revolving the region between x2+1x^2+1 and 3−x3-x around the xx-axis.

We begin by finding the intersection points of the curves. Setting x2+1=3−xx^2+1=3−x, we get 0=x2+x−2=(x−1)(x+2)0=x^2+x−2=(x−1)(x+2), and so the intersection points are x=−2x=−2 and x=1x=1.
We have that on [−2,4][−2,4], 3−x≥x2+13−x\ge x^2+1, so the volume of the solid is given by
V=π∫−21[3−x]2−[x2+1]2dx\displaystyle V = \pi\int^1_{−2}[3 − x]^2 − [x^2 + 1]^2 dx

=π∫−21(9−6x+x2−x4−2x2−1)dx\displaystyle= \pi\int^1_{−2}(9 − 6x + x^2 − x^4 − 2x^2 − 1)dx

=π∫−21(−x4−x2−6x+8)dx\displaystyle= \pi\int^1_{−2}(−x^4 − x^2 − 6x + 8) dx

=π(−x55−x33−3x2+8x)∣−21\displaystyle=\left.\pi\left(−\frac{x^5}{5} −\frac{x^3}{3} − 3x^2 + 8x\right)\right|^1_{−2}

=π[(−15−13−3+8)−(325+83−12−16)]\displaystyle= \pi\left[\left(−\frac{1}{5} −\frac{1}{3} − 3 + 8\right)−\left(\frac{32}{5}+\frac{8}{3} − 12 − 16\right)\right]

=π(−335−3+5+28)\displaystyle= \pi\left(−\frac{33}{5} − 3 + 5 + 28\right)

=π(150−335)\displaystyle= \pi\left(\frac{150 − 33}{5}\right)

=1175π\displaystyle=\frac{117}{5}\pi

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Example: Volumes of Revolution

Find the volume of a solid ball with radius aa .

The ball can be generated by rotating a half-disk. Since the radius is aa , we can use the half-disk whose area is generated by 0≤y≤a2−x20 \leq y \leq \sqrt{a^2 - x^2} where −a≤x≤a-a \leq x \leq a , and revolve it around the x-axis.


Note that visually, you could use a vertical half-disk instead, but that would no longer be a function of xx . A computation in this case can still be done, but it's easier, and more natural to do it in the way suggested initially. The volume then becomes
V=π∫−aa(a2−x2)2dx=2π∫0a(a2−x2)dxV=\pi\int^a_{-a}(\sqrt{a^2-x^2})^2dx=2\pi\int^a_0(a^2-x^2)dx

=2π(a2x−x33)∣0a=4πa33= \left . 2\pi\left(a^2x-\frac{x^3}3\right) \right |^a_0=\frac{4\pi a^3}{3}

Practice: Volumes of Revolution

The volume of the solid that is produced by revolving the region bounded between y=x2y=x^2 and y=xy=\sqrt x about the x-axis.

Practice: Volumes of Revolution

Write an integral representing the volume of the solid that is produced by revolving the region bounded between y=xy=x and y=3x−x2y=3x-x^2 about the line y=−1y=-1. Do not compute the integral.
Extra Practice
Volumes of Revolution by Cross Sections
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=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=2xy=\frac{2}{\sqrt{x}}, y=1y=1, and x=1x=1
a.) around the xx-axis
b.) around the yy-axis
Write the integrals representing the volume of the solid that is produced by revolving the region bounded between y=tan⁡3xy=\tan^3x, y=1y=1, and x=0x=0 about the line y=2y=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=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x} about the x-axis, between 0 and π/2\pi/2.
Write an integral representing the volume of the solid that is produced by revolving the region bounded between y=xy=x and y=3x−x2y=3x-x^2 about the line y=−1y=-1. Do not compute the integral.
Find the volume of the solid that is produced by revolving y=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x} about the xx-axis, between 0 and π/2\pi/2.
Find the volume of the solid that is produced by revolving y=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x} about the xx-axis, between 0 and π/2\pi/2.
Find the volume of the solid that is produced by revolving y=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x} about the xx-axis, between 0 and π/2\pi/2.
Let RR be the region in the first quadrant bounded by the graph of y=exy = e^x , the line y=2y = 2 and the yy-axis; let SS be the solid of revolution obtained by revolving RR about the xx-axis. Compute the volume of SS 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−x2y=5-x^2 and y=4y=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=2xy=\frac{2}{\sqrt{x}}, y=1y=1, and x=1x=1
a.) around the xx-axis
b.) around the yy-axis
Q:\bf{Q:} Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line y=2y=2.
Find the volume of the solid obtained by rotating the region bounded by the curves y=exy=e^x , y=1y=1 and x=4x=4, about the xx-axis and then about the yy-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=x3y=\sqrt[3]{x} and y=x4y=\dfrac{x}{4} that lies in the first quadrant, about the yy-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−x2y=5-x^2 and y=4y=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=2xy=\frac{2}{\sqrt{x}}, y=1y=1, and x=1x=1
a.) around the xx-axis
b.) around the yy-axis
Write an integral representing the volume of the solid that is produced by revolving the region bounded between y=xy=x and y=3x−x2y=3x-x^2 about the line y=−1y=-1. Do not compute the integral.
Practice: Volume of Cone
Q:\textbf{Q:} Derive the formula for the volume of a right circular cone whose altitude is hh and whose radius of the base is rr.
Practice: Rotate about Horizontal Line
Q:\textbf{Q:} Find the volume of the solid generated when the region enclosed by the curves y=x2y = x^2 and y=x+2y = x + 2 is revolved about the line y=−1. y = −1.
Practice: Rotate about y-axis
Q:\textbf{Q:} Consider the region RR of the first quadrant bounded by the curves y2=xy^2 = x and y=x4y = x^4. Find the volume of the region created by revolving R R around the y-axis. Use the method of cross-sections.
Practice: Rotate about x-axis
Q:\textbf{Q:} Find the volume of the solid obtained by revolving the region enclosed by y=4−5x22y=4-\dfrac{5x^2}{2} and y=x22+1y=\dfrac{x^2}{2}+1 around the xx-axis.
Practice: Rotate about x-axis
Q:\textbf{Q:} Find the volume of the solid obtained by revolving the region between y=x2+1y=x^2+1 and y=3−xy=3-x around the xx-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=2xy=\frac{2}{\sqrt{x}}, y=1y=1, and x=1x=1
a.) around the xx-axis
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=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−x2y=5-x^2 and y=4y=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−x2y=5-x^2 and y=4y=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−x2y=5-x^2 and y=4y=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−x2y=5-x^2 and y=4y=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−x2y=5-x^2 and y=4y=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−x2y=5-x^2 and y=4y=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−x2y=5-x^2 and y=4y=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−x2y=5-x^2 and y=4y=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−x2y=5-x^2 and y=4y=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−x2y=5-x^2 and y=4y=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=xy=\sqrt{x} and x=3yx=3y around the 𝑦-axis.
volume around y
Find the volume of the solid generated by rotating the region bounded by y=x2y=x^2 and y=xy=x about the yy-axis.
(a) π3\frac{\pi}{3} (b) π12\frac{\pi}{12} (c) π2\frac{\pi}{2} (d) 3π4\frac{3\pi}{4} (e) π6\frac{\pi}{6}
Write the integrals representing the volume of the solid that is produced by revolving the region bounded between y=tan⁡3xy=\tan^3x, y=1y=1, and x=0x=0 about the line y=2y=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=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x} about the x-axis, between 0 and π/2\pi/2.
Write an integral representing the volume of the solid that is produced by revolving the region bounded between y=xy=x and y=3x−x2y=3x-x^2 about the line y=−1y=-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=ex,  y=1,  x=ey=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=x3y=\sqrt[3]{x} and y=x4y=\dfrac{x}{4} that lies in the first quadrant, about the yy-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⁡3xy=\tan^3x, y=1y=1, and x=0x=0 about the line y=2y=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=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x} about the xx-axis, between 0 and π/2\pi/2.
Write an integral representing the volume of the solid that is produced by revolving the region bounded between y=xy=x and y=3x−x2y=3x-x^2 about the line y=−1y=-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=xy=x and y=3x−x2y=3x-x^2 about the line y=−1y=-1. Do not compute the integral.
Find the volume of the solid obtained by rotating the region bounded by x=5/yx=5/y , y=1y=1 and x=2x=2 about the y-axis.
volumeint
Write the integral that represents the volume of the solid obtained by rotating the region bounded by y=5xy=\frac{5}{x} , y=1y=1 and x=2x=2 about the x-axis.
Write the integral that represents the volume of the solid generated by revolving the region bounded by y=x3y=\sqrt[3]{x} and y=x4y=\dfrac{x}{4} that lies in the first quadrant, about the yy-axis. Use the method of disks/washers. Do not solve.
Q:\bf{Q:} Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line y=2y=2.
Find the volume of the solid obtained by rotating the region bounded by the curves y=exy=e^x , y=1y=1 and x=4x=4, about the xx-axis and then about the yy-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:} Find the volume of the solid obtained by revolving the region between y=x2+1y=x^2+1 and y=3−xy=3-x around the xx-axis.
Practice: Rotate about Horizontal Line
Q:\textbf{Q:} Find the volume of the solid generated when the region enclosed by the curves y=x2y = x^2 and y=x+2y = x + 2 is revolved about the line y=−1. y = −1.
Practice: Rotate about y-axis
Q:\textbf{Q:} Consider the region RR of the first quadrant bounded by the curves y2=xy^2 = x and y=x4y = x^4. Find the volume of the region created by revolving R R around the y-axis. Use the method of cross-sections.
Practice: Volume of Cone
Q:\textbf{Q:} Derive the formula for the volume of a right circular cone whose altitude is hh and whose radius of the base is rr.
Practice: Rotate about x-axis
Q:\textbf{Q:} Find the volume of the solid obtained by revolving the region enclosed by y=4−5x22y=4-\dfrac{5x^2}{2} and y=x22+1y=\dfrac{x^2}{2}+1 around the xx-axis.
Write the integrals representing the volume of the solid that is produced by revolving the region bounded between y=exy=e^x and y=(e−1)x+1y=(e-1)x+1 about the line y=−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=xy=x and y=3x−x2y=3x-x^2 about the line y=−1y=-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⁡3xy=\tan^3x, y=1y=1, and x=0x=0 about the line y=2y=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)=x2+1f\left(x\right)=x^2+1 , x=0x=0, x=2x=2 , and y=0y=0 is rotated around the y-axis.
Let RR be the region in the first quadrant bounded by the graph of y=exy = e^x , the line y=2y = 2 and the yy-axis; let SS be the solid of revolution obtained by revolving RR about the xx-axis. Compute the volume of SS 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 =3x3\sqrt{x} and 𝑦 = x4\frac{x}{4} 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=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x} about the xx-axis, between 0 and π/2\pi/2.
Volumes of Revolution by Cross Sections
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=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=2xy=\frac{2}{\sqrt{x}}, y=1y=1, and x=1x=1
a.) around the xx-axis
b.) around the yy-axis
volume around y
Find the volume of the solid obtained by rotating the region bounded by the curves y=xy=\sqrt{x} and x=3yx=3y around the 𝑦-axis.
Volumes of Revolution by Cross Sections
Find the volume of the solid generated by rotating the region bounded by y=x2y=x^2 and y=xy=x about the yy-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+2y=x+2 , y=x2y=x^2 andx=0x=0 around the y-axis.