Essential Concepts
- Definite integrals can be used to find the volumes of solids. Using the slicing method, we can find a volume by integrating the cross-sectional area.
- For solids of revolution, the volume slices are often disks and the cross-sections are circles. The method of disks involves applying the method of slicing in the particular case in which the cross-sections are circles, and using the formula for the area of a circle.
- If a solid of revolution has a cavity in the center, the volume slices are washers. With the method of washers, the area of the inner circle is subtracted from the area of the outer circle before integrating.
Key Equations
- Disk Method along the [latex]x[/latex]-axis
[latex]V={\displaystyle\int }_{a}^{b}\pi {\left[f(x)\right]}^{2}dx[/latex] - Disk Method along the [latex]y[/latex]-axis
[latex]V={\displaystyle\int }_{c}^{d}\pi {\left[g(y)\right]}^{2}dy[/latex] - Washer Method
[latex]V={\displaystyle\int }_{a}^{b}\pi \left[{(f(x))}^{2}-{(g(x))}^{2}\right]dx[/latex]
Glossary
- cross-section
- the intersection of a plane and a solid object
- disk method
- a special case of the slicing method used with solids of revolution when the slices are disks
- slicing method
- a method of calculating the volume of a solid that involves cutting the solid into pieces, estimating the volume of each piece, then adding these estimates to arrive at an estimate of the total volume; as the number of slices goes to infinity, this estimate becomes an integral that gives the exact value of the volume
- solid of revolution
- a solid generated by revolving a region in a plane around a line in that plane
- washer method
- a special case of the slicing method used with solids of revolution when the slices are washers
