Use the cylindrical shell method to find the volume of the solid generated by revolving the area bounded by the given curves

(x-3)^2 + y^2 = 9, about y-axis.

We are rotating a circle with centre at (3,0) and radius 3 around the y-axis

Because of symmetry we can use the part only above the x-axis, then double it.

see Khan for a nice explanation of the shell method
https://www.khanacademy.org/math/integral-calculus/solid-revolution-topic/shell-method/v/shell-method-for-rotating-around-vertical-line

by the shell method:
V = 2π[integral] x f(x) dx from 0 to 6

y^2 = 9 - (x-3)^2
= 9 - x^2+ 6x -9
= 6x - x^2
y = ± √(6x-x^2)
using only the upper part
y = f(x) = √(6x-x^2)

volume of upper part
= 2π [integral] x√(6x -x^2) dx from 0 to 6

nasty integral, Wolfram shows this:

http://www.wolframalpha.com/input/?i=integral+x%E2%88%9A(6x+-x%5E2)+from+0+to+6

As a check, the theorem of Pappus says that the volume is the area times the distance traveled by the centroid, which is 9π*6π = 54π^2.

To find the volume of the solid generated by revolving the area bounded by the given curves using the cylindrical shell method, you can follow these steps:

1. Understand the problem: The equation (x-3)^2 + y^2 = 9 represents a circle with a radius of 3 centered at (3, 0). We need to revolve this area around the y-axis.

2. Identify the limits of integration: Since we are revolving around the y-axis, the limits of integration will be the y-values where the two curves intersect. By solving the equation (x-3)^2 + y^2 = 9 for x, we get x = 3 ± √(9 - y^2). Thus, the limits of integration are y = -3 and y = 3.

3. Set up the integral: The volume of each cylindrical shell is given by the formula V = 2πrh * Δy, where r is the distance between the axis of rotation (in this case, the y-axis) and the shell, h is the height of the shell, and Δy is the change in y.

4. Calculate the radius and height: The radius of each cylindrical shell, r, is the distance from the axis of rotation (y-axis) to the curve (circle). Therefore, r = x. The height of each shell, h, can be calculated as h = Δx, where Δx is the change in x corresponding to the change in y, which can be found by rearranging the equation (x-3)^2 + y^2 = 9 as x = 3 ± √(9 - y^2).

5. Set up and solve the integral: The volume of the solid can be calculated by integrating the expression 2πrh * Δy from the lower limit to the upper limit. So, the integral for the cylindrical shell method is given by:

V = ∫[a,b] 2πrh * dy,

where r = x = 3 ± √(9 - y^2) and h = √(x^2) = √((3 ± √(9 - y^2))^2).

Evaluating this integral will give you the volume of the solid generated by revolving the area bounded by the given curves around the y-axis.