Solids of revolution? Find the volume of the region bounded by...?

Question

the region is bounded by the equations y = sqrt(x) y = 0 x = 4

The axis of revolution is x = 4

I know what the answer is, I just can't figure out how to write the integral.

I'm not asking anyone to solve the problem, if you could just show, or explain how to write the integral that is needed to solve.

Thanks in advance

Answer 1

You want to do this using cylindrical shells.  Draw the region and then rotate it around the line x=4.  Now draw in a typical cylindrical shell.  The center of the shell is the line x=4.  The height of the shell is given by the function y=sqrt(x), and the radius of the shell is the distance from the center to the shell, which is given by 4-x.  

If you approximated this using a finite number of shells, the thickness of each shell would be delta-x.  SInce we're using an integral, the thickness of the shell is dx.

Now the formula for the volume of the shell is 2(pi)(radius)(height)(thickness).  We need to integrate this over all the shells, which start at 0 and go to 4, so the volume is given by

∫ 2(pi) (4-x) (sqrt(x) dx

with lower endpoint 0 and upper endpoint 4.