1. Aim: Shell Method for Finding Volume Course: Calculus Do Now: Aim: How do we find volume using the Shell Method? Find the volume of the solid that results when the region bounded by x = y 3 and x = y 2 from y = 0 to y = 1 is revolved about the y-axis.

2. Aim: Shell Method for Finding Volume Course: Calculus The Shell Revolution w h pp Axis of Revolution h

3. Aim: Shell Method for Finding Volume Course: Calculus The Shell Method w h Outer radiusInner radius

4. Aim: Shell Method for Finding Volume Course: Calculus The Shell Method take to limit outer radius inner radius

5. Aim: Shell Method for Finding Volume Course: Calculus To find the volume of a solid of revolution with the shell method The Shell Method Horizontal Axis of Revolution c d h(y)h(y) Vertical Axis of Revolution h(x)h(x) ab p(x)p(x)

6. Aim: Shell Method for Finding Volume Course: Calculus Model Problem Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x 3 and the x-axis (0 < x < 1) about the y-axis. p(x)p(x) = x h(x)h(x) = x – x 3 Vertical Axis of Revolution

7. Aim: Shell Method for Finding Volume Course: Calculus Model Problem Find the volume of the solid of revolution formed by revolving the region bounded by and the y-axis (0 < y < 1) about the x-axis. h(y)h(y) p(y)p(y) Horizontal Axis of Revolution

8. Aim: Shell Method for Finding Volume Course: Calculus Comparing Disc and Shell Methods Disc Method Shell Method R c d r a b p h c d p h ab R r

9. Aim: Shell Method for Finding Volume Course: Calculus Do Now: Aim: How do we find volume using the Shell Method? The bases of a solid is the region enclosed by the graphs of y = 1/2x 2 and y = 8. Cross sections perpendicular to the y-axis are semicircles with diameter in the plane of the region. Find the volume of the solid. diameter

10. Aim: Shell Method for Finding Volume Course: Calculus Do-Now

11. Aim: Shell Method for Finding Volume Course: Calculus Recall: Model Problem disk washer disk washer

12. Aim: Shell Method for Finding Volume Course: Calculus Shell Method is Preferable p(x)p(x) h(x)h(x) = x 2 + 1 = x Vertical Axis of Revolution

13. Aim: Shell Method for Finding Volume Course: Calculus Model Problem A pontoon is to be made and is designed by rotating the graph of about the x-axis, where x and y are measured in feet. Find the volume. 8 ft. 2 ft. Disk method  a. rev Shell method  a. rev

14. Aim: Shell Method for Finding Volume Course: Calculus A pontoon is to be made and is designed by rotating the graph of about the x-axis, where x and y are measured in feet. Find the volume. Model Problem – Disk Method R(x) = r(x) = 0

15. Aim: Shell Method for Finding Volume Course: Calculus A pontoon is to be made and is designed by rotating the graph of about the x-axis, where x and y are measured in feet. Find the volume. Model Problem – Shell Method p(y) = y h(y)h(y) Horizontal Axis of Revolution 2( ) = h(y)

16. Aim: Shell Method for Finding Volume Course: Calculus Model Problem – Only Shell Find the volume of the solid of revolution formed by revolving the region bounded by the graphs of y = x 3 + x + 1, y = 1, x = 1 about the line x = 2. Axis of Revolution = 2 – x p(x)p(x) (1, 3) - ( 1) can’t solve for x

17. Aim: Shell Method for Finding Volume Course: Calculus Model Problem – Only Shell Find the volume of the solid of revolution formed by revolving the region bounded by the graphs of y = x 3 + x + 1, y = 1, x = 1 about the line x = 2. Axis of Revolution p(x)p(x) = 2 – x (1, 3) - ( 1) Vertical Axis of Revolution can’t solve for x

18. Aim: Shell Method for Finding Volume Course: Calculus The Shell Method