1. 6.4 Arc Length and Surface of Revolution
(Photo not taken by Vickie Kelly)
Greg Kelly, Hanford High School, Richland, Washington

2. Objectives Find the arc length of a smooth curve.
Find the area of a surface of revolution.

3. Rectifiable curve: One that has a finite arc length
f is rectifiable on [a,b] if f ' is continuous on [a,b].
If rectifiable, f is continuously differentiable on [a,b] and its graph is a smooth curve.

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

5. Length of Curve (Cartesian)
(function of x)
Length of Curve (Cartesian)
(function of y)

6. Example: Find the arc length of:

7. Example Find the arc length of:

8. Example: Find the arc length of:
Solve for x:
When x=0:
When x=8:

9. Example: Find the arc length of:

10. r Surface Area: Consider a curve rotated about the x-axis:
The surface area of this band is: r
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!

11. If revolving f(x) about x-axis
or g(y) about the y-axis:
r(x)=f(x)
r(y)=f(y)

12. Example:
Find the area of the surface formed by revolving
on [0,1] about the x-axis.
r=y

13. If revolving f(x) about y-axis
or g(y) about the x-axis:
r(x)=x
r(y)=y

14. Example:
Find the area of the surface formed by revolving
on about the y-axis.
r=x

15. (Use the calculator to evaluate integrals.)
Homework
6.4 (page 447)
#3-19 odd,
33-37 odd,
(Don't graph)
(Use the calculator to evaluate integrals.) p