1. 7.4 Lengths of Curves

2. 2+x csc x 1 0

3. If we want to approximate the length of a curve, over a short distance we could measure a straight line. By the pythagorean theorem: We need to get dx out from under the radical. Length of Curve (Cartesian)Lengths of Curves:

4. Example: Now what? This doesn’t fit any formula, and we started with a pretty simple example! The TI-89 gets:

5. Example: The curve should be a little longer than the straight line, so our answer seems reasonable. If we check the length of a straight line:

6. Example: You may want to let the calculator find the derivative too: Important: You must delete the variable y when you are done! ENTER F44 Y STO Y

7. Example:

8. If you have an equation that is easier to solve for x than for y, the length of the curve can be found the same way. Notice that x and y are reversed. ENTER X STO

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

10. Getting Around a Corner Find the length of the curve y = x 2 – 4|x| - x from x = -4 to x=4.

11. Ch 7.4 Surface Area

12. 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!

13. Example: Rotate about the y -axis.

14. Example: Rotate about the y -axis.

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

16. Example: rotated about x -axis. ENTER Y STO

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

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

19. 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.

20. Find the area of the surface formed by revolving the graph of f(x) = x 2 on the interval [0,√2] about the y axis.