# Volume integration

1. Feb 25, 2005

Calculate the volume generated when the region bounded by the curve y = 9 - X^2 , the line X = 2 and the x-axis is rotated 2pi about the y-axis.

For the part that lies in the second quadrant, i can integrate 9 to 0, then plug it into the volume formula. Then i have to add the volume of the part in the first quadrant. But since the X = 2 line slices a part of the area off i dunno how to integrate it respective to the y axis.

2. Feb 25, 2005

### MathStudent

Try using the method of cylindrical shells which should be outlined in your book.

Graph the region that is described. This is simple to do,, draw a downward porabola with y intercept at y=9, and a vertical line at x=2. The region you want to consider is the region in the first quadrant that lies under the porabola and over the x axis and between the lines x=2 and x=3.

hint:
the height of a shell will be 9 - x^2
the radius of the shell is x, so the circumference is 2(pi)x
the thickness of the shell is dx
you need to sum up all the shells with radius from x=2 to x=3

I hope I haven't given too much away.

Last edited: Feb 25, 2005
3. Feb 25, 2005

### BobG

Actually, you want the interval from 2 to 3.

The region would have to also be bounded by x=0 if you were going from 0 to 2.

4. Feb 25, 2005

### MathStudent

Yes, thank you BobG! I overlooked that.

P.S. I will edit my post so as not to cause confusion.