Shell Method Example: Finding Volume Using Shell Method

[Complexity]

Homework Statement

The area between the y-axis and the line y = x + 1 from y = 2 to
y = 4 is rotated about the x-axis. Use the shell method to find the
area of the resulting solid.

Homework Equations

V = 2pi integral (radius)(height of shell) dx or dy

The Attempt at a Solution

Well I am kind of confused with this problem, it says to find the AREA but isn't the shell method suppose to find the VOLUME?

Here is my attempt:

y= x+1 -----> x=y-1

V = 2pi integral from 2 to 4 y(y-1) dy

I then evaluated it to get = (76/3)pi as the volume.

What did I do wrong?

 

[Dick]

I'll agree with what you've done so far. The volume is 76*pi/3. If they really do want area I'd add up the area of the parts of the boundary. You've got two cylinders, a disk with a hole in it and a part of a cone. It seems like it shouldn't be too hard, but I don't know what that has to do with the 'shell method'.