5. Find the volume generated by rotating the region bounded curve

y = (1-x)(3-x), 1 <= x <= 3 and x-axis from 1 to 3

(a)around the y-axis. (b) around the line x = 2

Solution:

Start with the curve y = (1 - x)(3 -x)

 

 

Since the function is negative on its domain the integrals will have to be multiplied by -1 to get the volume.

 

(a) Rotating around the y-axis, the radius of the shell is x and the height is (x-1)(3-x). (Changing 1-x to x-1 changes the sign for x >= 1.)

So the volume is equal to

(b) We shall use the shell method, an rotate about the line x = 2. The radius of a typical disk is (2-x),and the height is -y = (x-1)(3-x)so the corresponding volume is

Error you might make here: Since you are using the shell method, and since the figure is symmetric about the line x = 2, the range of integration is 1 to 2.