1. Lecture 4 – Arc Length Arc Length – curve has a certain length 1 2 1 Estimate the length of the given through appropriate choices for lower and upper bounds. 3 4 1 2 3 4

2. Lengths (in general) Use more and more hypotenuse lengths to better the approximations. 2 f(x) x Consider one segment out of n pieces and then let n go to infinity.

3. Example 1 Find the length of the given curve. 3 1 2 3 4

4. Example 2 Find the length of the given curve. 4 f(x) x 12 1

5. Example 3 Find the length of the given curve. 5 f(x) x 2 1

6. Example 3 – continued 6 f(x) x 2 1

7. Lecture 5 – Surface Area 7 Find the surface area by revolving the curve around the y-axis.

8. 8 Find the surface area by revolving the curve around the x-axis.

9. Example 1 Put the surface areas in decreasing order for the given curve. 9 A: revolved about the y-axis. B: revolved about the x-axis. C: revolved about the line x=2.

10. 10 Example 1-continued 1 2 3 4 1 2 3 4 1 2 3 4