Applications of integration
Volumes of revolution
Rotate a region around a stated axis. Choose disks, washers, or cylindrical shells and integrate their volume contributions.
1. Disks around the x-axis
1 / 3V=\pi\int_0^b x^2dx=\frac{\pi b^3}{3} A cone formed by rotating y=x about the x-axis.
Rotate the region under y=x from 0 to b about the x-axis. A disk at x has radius x and area πx². Integrating gives the cone volume πb³/3.
- radius at b
- 1
- V/π
- 0.333333
- V
- 1.047198
Switch between the generating region and the solid of revolution. Rotate the view to inspect the shape; the dashed line is the rotation axis. Changing the view does not change the volume.
2. Washers leave a hole
2 / 3V=\pi\int_0^b(2^2-1^2)dx=3\pi b A hollow cylinder with outer radius 2 and inner radius 1, around the x-axis.
Rotate 1≤y≤2, 0≤x≤b about the x-axis. Each washer has outer radius 2 and inner radius 1. Subtract the squared radii before integrating.
- R
- 2
- r
- 1
- V/π
- 3
Switch between the generating region and the solid of revolution. Rotate the view to inspect the shape; the dashed line is the rotation axis. Changing the view does not change the volume.
3. Shells around the y-axis
3 / 3V=2\pi\int_0^b x\,dx=\pi b^2 A cylinder of height 1 and radius b, around the y-axis.
Rotate 0≤y≤1, 0≤x≤b about the y-axis. A shell at x has circumference 2πx and height 1. The integral gives πb², the volume of a cylinder of height 1.
- radius
- 1
- height
- 1
- V/π
- 1
Switch between the generating region and the solid of revolution. Rotate the view to inspect the shape; the dashed line is the rotation axis. Changing the view does not change the volume.