1. Warm Up

2. Volume of Solids - 8.3A

3. Big Idea Just like we estimate area by drawing rectangles, we can estimate volume by cutting the shape into slices, finding the area of each slice, and adding them together.

4. Definition: Volume of a Solid The volume of a solid with known integrable cross section area A(x) from x = a to x = b is

5. 1 Find a formula for A(x) Sketch the solid and a typical cross section. 2 3 Find the limits of integration. 4 Integrate A(x) to find volume V(x) Steps

6. E XAMPLE A pyramid 3 meters high has a square base that is 3 m on each side. Each cross section parallel to the base is square. Find the volume of the pyramid. 3 3

7. This correlates with the formula: Solution

8. Solid of revolution If we take a 2-D shape and rotate it around an axis, we end up with a 3D shape that we can find the volume of. The only thing that changes when the cross-sections are circular is the formula for A(x).

9. Example

11. Solution

12. Homework 8.3A Skip 17