Question

How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region `y=2-x`, `2<=x<=4` rotated about the x-axis?

Answer 1

Always graph the constraints first, then determine the correct integration formula.

Explanation:

Here is a graph of the constraints:

Now, without using calculus, you should visualize rotating this line around the x-axis will generate a cone with radius 2 and height 2. The volume of a cone is:

`V=(1/3)pir^2h=(1/3)pi2^2*2= (8/3)pi`

So, in advance of using calculus, we know the answer is `(8/3)pi`

Now, using calculus and the shell method :

Because we rotate around the x-axis, integrate with respect to y:

`2piint_-2^0y[4-(2-y)]dy=(8/3)pi`

hope that helped