Calculus · Quick Reference

Volumes Cheat Sheet

Integral Calculus · Lesson 13/20

In one line: integration extends to three dimensions, letting us calculate the volume of solids formed by rotating or stacking two-dimensional regions.

Key Ideas

1Disk Method. When a region is rotated around an axis and there's no gap, the volume is Integral from a to b of pi*[f(x)]^2 dx.

2Washer Method. When rotating a region between two curves (leaving a hole), Volume = Integral from a to b of pi*([outer(x)]^2 - [inner(x)]^2) dx.

3Choosing the Axis of Rotation. Rotating around the x-axis uses functions of x; rotating around the y-axis requires rewriting functions in terms of y.

4Cross-Sectional Method. More generally, Volume = Integral from a to b of A(x) dx, where A(x) is the area of the cross-section at position x.

5Units. If lengths are in cm, area is in cm^2, and volume comes out in cm^3.

Worked Examples

Find the volume of the solid formed by rotating f(x)=x from x=0 to x=2 around the x-axis.

V = 8pi/3 cubic units

Find the volume formed by rotating f(x) = sqrt(x) from x=0 to x=4 around the x-axis.

V = 8pi cubic units

Find the volume between R(x)=x+1 (outer) and r(x)=1 (inner) rotated around the x-axis from x=0 to x=2, using the washer method.

V = 20pi/3 cubic units

Formulas

Disk method: V = Integral from a to b of pi*[f(x)]^2 dx

Washer method: V = Integral from a to b of pi*([R(x)]^2 - [r(x)]^2) dx

Practice Yourself

Find the volume rotating f(x)=x from x=0 to x=3 (disk method).

9pi

Find the volume rotating f(x)=2 (constant) from x=0 to x=5 (disk method).

20pi

Find the volume rotating f(x)=sqrt(x) from x=0 to x=1 (disk method).

pi/2