1. 6.3 – Volumes of Cylindrical Shells

2. Derivation Assume you have a functions similar to the one shown below and assume the f is to difficult to solve for y in terms of x. Rotate f about the y axis. y = f (x) xixi xoxo It is too difficult to find x i and x o so that we can find the area of the washer. We need another method for more complicated functions.

3. Cylindrical Shells Let x i be some subinterval and establish a rectangle to estimate the area. When this rectangle is rotated about the y-axis, a cylindrical shell if formed. y = f (x) xixi hr2r2 ΔrΔr r1r1

4. Cylindrical Shells Now determine the volume of the cylindrical shell. V 2 is the volume of the larger right circular cylinder and V 1 is the volume of the inner. hr2r2 ΔrΔr r1r1

6. Cylindrical Shells Now assume that the interval [a, b] is divided into n subintervals of equal width Δx. Also assume that x i is taken at the mid-point of each subinterval. This means that r i = x i the height of each cylindrical shell is given by f (x i ). The volume of each cylindrical shell is

7. Cylindrical Shells An estimate of the total volume is the sum of the volumes of the n cylindrical shells.

8. Volume of a Solid Using Cylindrical Shells Therefore, the volume of the solid obtained by rotating about the y-axis the region under the curve y = f (x) from a to b is

9. Volume of a Solid Using Cylindrical Shells Assume you have a functions similar to the one shown below and assume the f is to difficult to solve for y in terms of x. Rotate f about the y axis.