1. 7.4 Day 2 Surface Area Greg Kelly, Hanford High School, Richland, Washington(Photo not taken by Vickie Kelly)

2. Surface Area: r Consider a curve rotated about the x -axis: The surface area of this band is: The radius is the y -value of the function, so the whole area is given by: This is the same ds that we had in the “length of curve” formula, so the formula becomes: Surface Area about x -axis (Cartesian): To rotate about the y -axis, just reverse x and y in the formula!

3. Surface Area: r Difference between surface area and shell method: With surface area, we are integrating with respect to the change is s (ds) so summing up the arc length which is the height of the (shell) only. We find volumes with shell method because we are integrating the change in x (dx) so accumulating the thicknesses to find the volume.

4. Example: Rotate about the y -axis.

5. Example: Rotate about the y -axis.

6. Example: Rotate about the y -axis. From geometry:

7. Example: rotated about x -axis. The TI-89 gets:

8. Example: Check: rotated about x -axis. ENTER Y STO

9. Don’t forget to clear the x and y variables when you are done!  ENTER F44 Y X Once again …

10. Example: The Area of a Surface of Revolution Find the area of the surface formed by revolving the graph of f(x) = x 3 on the interval [0, 1] about the x-axis, as shown in Figure 7.46. Figure 7.46

11. Solution: The distance between the x-axis and the graph of f is r(x) = f(x), and because f'(x) = 3x 2, the surface area is

12. Example: The Area of a Surface of Revolution Find the area of the surface formed by revolving the graph of f(x) = x 2 on the interval [0, ] about the y-axis, as shown.

13. Solution: The Area of a Surface of Revolution In this case, the distance between the graph of f and the y-axis is Using the surface area is:

14. Homework: 7.2 day 2: MMM pgs. 59 & 62.