1. (SEC. 7.3 DAY ONE) Volumes of Revolution DISK METHOD

2. Example 1: Find the volume of a sphere with a radius of 2. METHOD 1: GEOMETRY!!!!!!!!!!

3. Example 1: Find the volume of a sphere with a radius of 2. METHOD 2: CALCULUS!!!!!!!!!! Step One: Write an equation to represent the edge of the shape (in this case: a CIRCLE!!!). Step Two: Solve the equation for y. What shape would result from rotating this “function” over the x-axis? A SPHERE!!! CIRCLE! Step Three: Determine what shape cross-sections (made perpendicular to x- axis) of the sphere are.

4. Step Five: SUM up the area of all the possible cross-sections. In calculus, we SUM using an INTEGRAL!!! Step Six: Evaluate the integral. Compare our answer above to the one we got using the geometry formula!! WE GET THE SAME ANSWER! CALCULUS WORKS!!! Step Four: Write the equation for the area of one of the cross-sections (in terms of x).

5. What shape is this problem referring to? A CYLINDER!!! Use GEOMETRY to find the volume of the cylinder. Use CALCULUS to find the volume of the cylinder.

6. What shape is this problem referring to? Use GEOMETRY to find the volume of the cylinder. Use CALCULUS to find the volume of the cylinder. A CONE!! Consider what shape one cross-section (taken perpendicular to the x-axis) of the solid would be. A CIRCLE!!

7. So what happens the solid is not one we have a geometric formula for? You have to use CALCULUS!!! Here’s the basic formula: Radius of circular cross-section

8. Evaluate this using your calculator…

10. Because the circular cross-sections will be horizontal, we will integrate this time with respect to y! This means the bounds for integration should be y-values and the function must be solved for x.

12. The key here is that you HAVE to use TWO integrals! We need to determine EXACTLY where the functions intersect first…

13. 1. TO ROTATE OVER A LINE OTHER THAN ONE OF THE AXES. 1. TO ROTATE AN AREA NOT FORMED BY ONE OF THE AXES. We still need to learn….

14. HOMEWORK