1. Section 7.3 – Volume: Shell Method

2. White Board Challenge Calculate the volume of the solid obtained by rotating the region bounded by y = x 2, x=0, and y=4 about x = -1. Calculator We will now investigate another method to calculate this volume.

3. Volume of a Shell Consider the following cylindrical shell (formerly a washer): r outer h r inner The average of the radii is a new radius from the center of the base to the middle of the enclosed area. R Imagine the circle in in the middle of the base area. Label the new radius. Thus, the circumference of the middle circle is… Also, the thickness of the shell is… ΔrΔr

4. Volume of a Shell The volume of the cylindrical shell is easier to see when it is flattened out: h C = 2πR The cylindrical shell flattened out is a rectangular prism. The length of the base is… The height of the base is… The height of the prism is… ΔrΔr Thus the volume of the prism is…

5. Volumes of Solids of Revolution with Riemann Sums Let us rotate the region under y=f(x) from x=a to x=b about the y - axis. The resulting solid can be divided into thin concentric shells.

6. Volumes of Solids of Revolution: Shell Method Sketch the bounded region and the line of revolution. If the line of revolution is horizontal, make sure the equations can easily be written in the x= form. If vertical, the equations must be in y= form. Sketch a generic shell (a typical cross section). Find the radius of the generic shell (perpendicular distance from the line revolution to the outer edge of the shell), the height of the shell (this length is perpendicular to the radius), and the thickness of the shell (this length is perpendicular to the height). Integrate with the following formula: Opposite of Washer Method MAKE A HOOK:

7. Example 1 Calculate the volume of the solid obtained by rotating the region bounded by y = x 2, x=0, and y=4 about x = -1. Sketch a Graph Find the Boundaries/Intersections Make Generic Shell(s) Height = 4 – x 2 Integrate the Volume of the Shell x We only need x>2 Line of Rotation Radius = x – -1 = x + 1 Thickness = dx

8. Example 2 Calculate the volume V of the solid obtained by rotating the region bounded by y = 5x – x 2 and y = 8 – 5x + x 2 about the line y -axis. Sketch a Graph Find the Boundaries/Intersections Make Generic Shell(s) Height = ( 5x – x 2 ) – (8 – 5x + x 2 ) Integrate the Volume of Each Generic Washer Line of Rotation Radius = x Thickness = dx

9. White Board Challenge Use the shell method to calculate the volume of the solid obtained by rotating the region bounded by y = x 1/2 and y=0 over [0,4] about the x -axis. Calculator Line of Rotation Radius = y Height = 4 – y 2 Thickness = dy