1. Arc Length and Surfaces of Revolution Lesson 7.4

2. What Is Happening?

3. Arc Length We seek the distance along the curve from f(a) to f(b)  That is from P 0 to P n The distance formula for each pair of points a b P0P0 P1P1 PnPn PiPi Why? What is another way of representing this?

4. Arc Length We sum the individual lengths When we take a limit of the above, we get the integral

5. Arc Length Find the length of the arc of the function for 1 < x < 2

6. Surface Area of a Cone Slant area of a cone Slant area of frustum s r h L

7. Surface Area Suppose we rotate the f(x) from slide 2 around the x-axis  A surface is formed  A slice gives a cone frustum a b P0P0 P1P1 PnPn PiPi xixi ΔsΔs ΔxΔx

8. Surface Area We add the cone frustum areas of all the slices  From a to b  Over entire length of the curve

9. Surface Area Consider the surface generated by the curve y 2 = 4x for 0 < x < 8 about the x-axis

10. Surface Area Surface area =

11. Limitations We are limited by what functions we can integrate Integration of the above expression is not trivial We will come back to applications of arc length and surface area as new integration techniques are learned

12. Assignment Lesson 7.4 Page 383 Exercises 1 – 29 odd also 37 and 55,