1. More on Volumes & Average Function Value

2. Average On the last test (2), the average of the test was: FYI - there were 35 who scored a 9 or 10, which means the median was a 9.

3. Average Function Value of f(x) on the interval [a,b] We can divide the interval [a,b] into n subintervals and average the selected function values.

4. Average function value If we let then number of points selected go to infinity We arrive at the Definite Integral!

5. Find the average function value of over the interval

7. Solving for x, we get: The area of the green rectangle = the area under over the interval

8. Average function value If we let multiply both sides of the formula, we get: Thinking area: The area of the rectangle (b-a) by Ave f has the same area as the area under the curve as seen…..

9. Let R be the region in the x-y plane bounded by Set up the integral to find the area of the region. Top Function: Bottom Function: Area: Bounds: [0,2] Length:

10. Set up the integral to find the volume of the solid whose base is the region between the curves: and with cross sections perpendicular to the x-axis equilateral triangles. Area : Volume: Length:

11. Find the volume of the solid generated by revolving the region defined by about the line y = 3. Area : Volume: Length: Inside (r) : Outside (R) :

12. Solids of Revolution If there is a gap between the function and the axis of rotation, we have a washer and use: If there is NO gap, we have a disk and use: Volume =

13. Note in the disk/washer methods, the focus in on the radius (perpendicular to the axis of rotation) and the shape it forms. We can also look at a slice that is parallel to the axis of rotation.

14. Area: Length of slice

15. Volume = Slice is PARALLEL to the AOR

16. Using on the interval [0,2] revolving around the x-axis using planar slices PARALLEL to the AOR, we find the volume: Radius? Length of slice? Area? Volume?

17. Back to example: Find volume of the solid generated by revolving the region about the y-axis using cylindrical slices Length of slice ( h ): Radius ( r ): Area: Volume:

18. Find the volume of the solid generated by revolving the region: about the y-axis, using cylindrical slices. Length of slice ( h ): Inside Radius ( r ): Area: Volume:

19. Find the volume of the solid generated by revolving the region defined by about the line x = 3. Area : Volume: Length: Radius :

20. Let R be the region in the x-y plane bounded by Set up the integral for the volume of the solid obtained by rotating R about the line y = 3, a) Integrating with respect to x. b) Integrating with respect to y.

21. Integrating with Respect to x: Outside Radius ( R ): Inside Radius ( r ): Area: Volume:

22. Integrating with Respect to y: Length of Slice: Inside Radius ( r ): Area: Volume: