Prerequisites: Riemann Sums, Definition of Definite Integral

Goal: To discover a method to evaluate the volume of a solid of rotation

1. How is the bottom left graph related to the top left graph? In particular, what concept does the bottom left graph demonstrate about the top left graph?
2. How is the top right graph related to the top left graph?
3. How is the bottom right graph related to the top right graph? In particular, what concept does the bottom right graph demonstrate about the top right graph?
4. Consider the i^th rectangle in the bottom left graph.

   a. If the midpoint of the interval is x_i^*, what is the height of the rectangle?

   b. What is its width? Hint: we had a short-hand notation for this.

   c. What is its area?

5. Write a Riemann Sum for the area of the bottom left graph. Use n for the number of sub-intervals.
6. Write a Definite Integral for the area of the top left graph.
7. Consider the i^th cylinder in the bottom right graph.

   a. If the midpoint of the interval is x_i, what is the radius of the cylinder?

   b. What is the area of a cross section of the cylinder?

   c. What is the thickness of the cylinder? Hint: we can recycle short-hand notations.

   d. What is its volume?

8. Write a Riemann Sum for the volume of the bottom right graph. Use n for the number of sub-intervals.
9. Write a Definite Integral for the volume of the top right graph.