1. Arc Length and Surfaces of Revolution
Lesson 7.4
Arc Length and Surfaces of Revolution

2. We want to be able to find the distance travelled by an object along a curve
This is the same as finding the length of a curve from a given point to another
The integral we will use comes from the infinite limit of the lengths of segments along the curve

3. Arc Length Along a Curve
Distance, s, is given by or
If the function is differentiable along the interval [a, b]

4. Example
Find the arc length on the interval [1, 2]

5. Area of a Surface of Revolution
We are trying to find the lateral area of the cylinder below:
The formula for this is found using circumference and “length” of the cylinder
circumference
Cylinder length

6. The length, L, in this formula is the length of the curve used earlier
Then, applying an infinite limit of infinitely small cylinders along with a changing radius, we get the integral for surface area of revolution:
If the length, L, of the cylinder is vertical then

7. Answer Example Find the surface area formed by revolving
around the x-axis from 1 to 4.
Answer
~30.85

8. Answer Example Find the surface area formed by revolving
around the y-axis from 0 to 2.
Answer
~36.18