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?

1 Answer
BeeFree
Oct 9, 2015

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

Explanation:

Here is a graph of the constraints:

enter image source here

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

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