Volume The Shell Method Lesson 7 3 Find

Volume: The Shell Method Lesson 7. 3

Find the volume generated when this shape is revolved about the y axis. We can’t solve for x, so we can’t use a horizontal slice directly.

If we take a vertical slice and revolve it about the y-axis we get a cylinder.

Shell Method • Based on finding volume of cylindrical shells § Add these volumes to get the total volume • Dimensions of the shell § Radius of the shell § Thickness of the shell § Height

The Shell • Consider the shell as one of many of a solid dx of revolution f(x) – g(x) x g(x) • The volume of the solid made of the sum of the shells

Try It Out! • Consider the region bounded by x = 0, y = 0, and

Hints for Shell Method • • Sketch the graph over the limits of integration Draw a typical shell parallel to the axis of revolution Determine radius, height, thickness of shell Volume of typical shell • Use integration formula

Rotation About x-Axis • Rotate the region bounded by y = 4 x and y = x 2 about the x-axis thickness = dy radius = y • What are the dimensions needed? § radius § height § thickness

Rotation About Non-coordinate Axis • Possible to rotate a region around any line g(x) f(x) x=a • Rely on the basic concept behind the shell method

Rotation About Non-coordinate Axis • What is the radius? r f(x) a–x • What is the height? f(x) – g(x) • What are the limits? c<x<a • The integral: x=c x=a g(x)

Try It Out • Rotate the region bounded by 4 – x 2 , x = 0 and, y = 0 about the line x = 2 r=2 -x 4 – x 2 • Determine radius, height, limits

Try It Out • Integral for the volume is

Assignment • • Lesson 7. 3 Page 472 Exercises 1 – 25 EOO Also … 27, 29, 35, 37, 41, 43, 55