01 Key Concepts
Disk 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.
Washer 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.
Choosing 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.
Cross-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.
Units
If lengths are in cm, area is in cm^2, and volume comes out in cm^3.
02 Key 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
03 Solved Examples
Example 1 Find the volume of the solid formed by rotating f(x)=x from x=0 to x=2 around the x-axis.
- Use the disk method: V = Integral from 0 to 2 of pi*x^2 dx.
- Antiderivative: pi*x^3/3.
- Evaluate: pi*(8/3) - 0.
Answer: V = 8pi/3 cubic units
Example 2 Find the volume formed by rotating f(x) = sqrt(x) from x=0 to x=4 around the x-axis.
- V = Integral from 0 to 4 of pi*(sqrt(x))^2 dx = Integral from 0 to 4 of pi*x dx.
- Antiderivative: pi*x^2/2.
- Evaluate: pi*(16/2) - 0.
Answer: V = 8pi cubic units
Example 3 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 = Integral from 0 to 2 of pi*[(x+1)^2 - 1^2] dx.
- Expand: (x+1)^2 - 1 = x^2 + 2x.
- Integrate: pi*(x^3/3 + x^2).
- Evaluate at x=2: pi*(8/3+4) = pi*(20/3). At x=0: 0.
Answer: V = 20pi/3 cubic units
04 Practice Questions
1Find the volume rotating f(x)=x from x=0 to x=3 (disk method).
9pi
2Find the volume rotating f(x)=2 (constant) from x=0 to x=5 (disk method).
20pi
3Find the volume rotating f(x)=sqrt(x) from x=0 to x=1 (disk method).
pi/2
4What method is used when there's a hole in the rotated solid?
The washer method
5If lengths are in meters, what unit is the resulting volume in?
Cubic meters
📄 Volumes — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.