1. Arc Length and Surface Area
Lesson 6.4

2. What Is Happening?

3. What is another way of representing this?
Arc Length
We seek the distance along the curve from f(a) to f(b)
That is from P0 to Pn
The distance formula for each pair of points P1 • Pi P0 Pn • • • • • b a
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 h

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 P1 Pi P0 Pn • • • • • • • xi b a Δs

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 y2 = 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 6.4
Page 393
Exerxises 1 – 25 odd