1. Section 7.3: Volume The Last One!!! Objective: Students will be able to… Find the volume of an object using one of the following methods: slicing, disk, washer

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

3. Cross Section Areas (images)

4. How to Find Volume by Slicing 1. Sketch the solid (or the base of the solid) and a typical cross section 2. Find a formula for A(x)[area of the cross section] 3. Find the limits of integration 4. Integrate A(x) to find the volume

5. To help with visualizing: https://www.youtube.com/watch?v=omQSp2uMYT k https://www.youtube.com/watch?v=omQSp2uMYT k

6. Example: Find the volume of the solid that lies between planes perpendicular to the x-axis and x = 0 and x = 4. The cross sections perpendicular to the x axis between these planes run from one side of the parabola x = y 2 to the other. The cross sections are squares with bases in the xy plane.

7. Example Work:

8. Helpful Applets…Play with them! Triangle Base: http://web.monroecc.edu/manila/webfiles/pseeburger/secure/MyLarson/ch 7/LC7_2xsection1.htm Circle Base: http://web.monroecc.edu/manila/webfiles/pseeburger/secure/MyLarson/ch 7/LC7_2xsection2.htm Solids in xy-plane: http://web.monroecc.edu/manila/webfiles/calcNSF/JavaCode/other/myXSe ction.htm

9. Find the volume of the solid that lies between planes perpendicular to the x axis at x = -1 and x = 1. The cross sections perpendicular to the x axis are circular disks whose diameters run from the parabola y = x 2 to the parabola y = 2-x 2.