Volume of Rotated Region Using Cylindrical Shells - Step-by-Step Guide

[shaneewert]

Please help! How do I do this problem?

Using the method of cylindrical shells, find the volume generated by rotating the region the region bounded by the given curves about the specified axis.

y=(x-1)^(1/2), y=0, x=5; about y = 3

Please tell me how to set up the integral! Any help is MUCH appreciated.

So far I have Integral from 0 to 2 of (3-(y^2+1))*y dy I know that isn't right, because I am not getting the right answer! The book says that it is 24pi.

Thank you.

 

[0rthodontist]

Do you know what each shell looks like?

The cross section of each shell is a long, narrow, flat rectangle, whose long dimension is parallel to the x-axis and stretches from the point where x = y^2 + 1 to the point where x = 5. It is parametrized by y. The shell itself is that rectangle rotated around the line y = 3. Ignore the small vertical width of the rectangle for a minute (which is dy) so that the shell is just like a horizontal tube with no thickness. What is the area of the outer surface of this tube? It has a radius--the radius is 3 - (y^2 + 1). So what is its circumference? What is its length?