Volumes of Revolution Formula

Volumes of revolution is finding the volume of a three-dimensional solid formed by rotating a two-dimensional region around an axis.

The Formula

Disc: V=Ο€βˆ«ab[f(x)]2 dxV = \pi\int_a^b [f(x)]^2\,dx
Washer: V=Ο€βˆ«ab([R(x)]2βˆ’[r(x)]2)dxV = \pi\int_a^b \left([R(x)]^2 - [r(x)]^2\right)dx (outer radius RR, inner radius rr)
Shell: V=2Ο€βˆ«abx f(x) dxV = 2\pi\int_a^b x\,f(x)\,dx (rotating around yy-axis)

When to use: Spin a flat region around a line, like spinning a pottery wheel. The flat shape sweeps out a 3D solid. To find its volume, slice the solid into thin pieces (discs, washers, or shells), find the volume of each slice, and add them upβ€”which means integrate.

Quick Example

Rotate f(x)=xf(x) = \sqrt{x} around the xx-axis from x=0x = 0 to x=4x = 4 (disc method):
V=Ο€βˆ«04(x)2 dx=Ο€βˆ«04x dx=Ο€[x22]04=8Ο€V = \pi\int_0^4 (\sqrt{x})^2\,dx = \pi\int_0^4 x\,dx = \pi\left[\frac{x^2}{2}\right]_0^4 = 8\pi

Notation

Disc = solid circular cross-section (no hole). Washer = annular cross-section (ring with a hole). Shell = thin cylindrical layer.

What This Formula Means

Finding the volume of a three-dimensional solid formed by rotating a two-dimensional region around an axis. The disc/washer method uses circular cross-sections perpendicular to the axis; the shell method uses cylindrical shells parallel to the axis.

Spin a flat region around a line, like spinning a pottery wheel. The flat shape sweeps out a 3D solid. To find its volume, slice the solid into thin pieces (discs, washers, or shells), find the volume of each slice, and add them upβ€”which means integrate.

Formal View

Disc: V=Ο€βˆ«ab[f(x)]2 dxV = \pi \int_a^b [f(x)]^2\,dx. Washer: V=Ο€βˆ«ab([R(x)]2βˆ’[r(x)]2)dxV = \pi \int_a^b \left([R(x)]^2 - [r(x)]^2\right)dx. Shell: V=2Ο€βˆ«abxβ‹…f(x) dxV = 2\pi \int_a^b x \cdot f(x)\,dx. All derived from V=∫abA(x) dxV = \int_a^b A(x)\,dx where A(x)A(x) is the cross-sectional area.

Worked Examples

Example 1

easy
Find the volume of the solid formed by rotating f(x)=xf(x) = \sqrt{x} around the xx-axis from x=0x=0 to x=4x=4.

Answer

V=8Ο€V = 8\pi

First step

1
Use the disc method for revolution about the xx-axis: V=Ο€βˆ«ab[f(x)]2 dxV = \pi\int_a^b [f(x)]^2\,dx. Since f(x)=xf(x) = \sqrt{x}, [f(x)]2=x[f(x)]^2 = x.

Full solution

  1. 2
    Set up the integral: V=Ο€βˆ«04x dxV = \pi\int_0^4 x\,dx
  2. 3
    Evaluate: V=Ο€[x22]04=Ο€β‹…162=8Ο€V = \pi\left[\frac{x^2}{2}\right]_0^4 = \pi\cdot\frac{16}{2} = 8\pi
The disc method stacks thin circular discs. Each has radius f(x)f(x) and thickness dxdx, giving volume element Ο€[f(x)]2 dx\pi[f(x)]^2\,dx.

Example 2

hard
Find the volume when the region between y=xy=x and y=x2y=x^2 (0≀x≀10 \leq x \leq 1) is rotated around the xx-axis.

Example 3

medium
Find the volume when y=x3y=x^3, 0≀x≀10\le x\le 1, is revolved about the xx-axis.

Common Mistakes

  • Forgetting the Ο€\pi in disc/washer formulas β€” cross-sections are circles of area Ο€r2\pi r^2.
  • Using a disc when a gap to the axis makes a hole β€” that requires the washer method (subtract inner radius squared).
  • Mismatching method to the slicing direction β€” discs/washers slice perpendicular to the axis, shells parallel; pick the one that makes radii simple.

Why This Formula Matters

This turns the area integral into a volume integral and is the capstone of intro calculus's geometric applications β€” modeling everything from a vase's volume to engineering solids. The decisive choice is the method: a solid disc (no hole), a washer (hole from a gap to the axis), or a shell (when slicing parallel to the axis is simpler); picking wrong makes the integral much harder. Recognizing it by "Is a flat region being spun around an axis to make a solid whose volume I find by integrating cross-sections?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from area between curves and disc vs washer vs shell and surface area of revolution in a mixed problem set.

Frequently Asked Questions

What is the Volumes of Revolution formula?

Finding the volume of a three-dimensional solid formed by rotating a two-dimensional region around an axis. The disc/washer method uses circular cross-sections perpendicular to the axis; the shell method uses cylindrical shells parallel to the axis.

How do you use the Volumes of Revolution formula?

Spin a flat region around a line, like spinning a pottery wheel. The flat shape sweeps out a 3D solid. To find its volume, slice the solid into thin pieces (discs, washers, or shells), find the volume of each slice, and add them upβ€”which means integrate.

What do the symbols mean in the Volumes of Revolution formula?

Disc = solid circular cross-section (no hole). Washer = annular cross-section (ring with a hole). Shell = thin cylindrical layer.

Why is the Volumes of Revolution formula important in Math?

This turns the area integral into a volume integral and is the capstone of intro calculus's geometric applications β€” modeling everything from a vase's volume to engineering solids. The decisive choice is the method: a solid disc (no hole), a washer (hole from a gap to the axis), or a shell (when slicing parallel to the axis is simpler); picking wrong makes the integral much harder. Recognizing it by "Is a flat region being spun around an axis to make a solid whose volume I find by integrating cross-sections?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from area between curves and disc vs washer vs shell and surface area of revolution in a mixed problem set.

What do students get wrong about Volumes of Revolution?

The procedure for volumes of revolution is the easy part; the trap is forgetting the Ο€\pi in disc/washer formulas. Asking "Is a flat region being spun around an axis to make a solid whose volume I find by integrating cross-sections?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Volumes of Revolution formula?

Before studying the Volumes of Revolution formula, you should understand: area between curves, definite integral.

Want the Full Guide?

This formula is covered in depth in our complete guide:

How to Integrate Rational Functions: Long Division and Partial Fractions β†’
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