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Integration to calculate Volume
By Lee Lucy on Friday, August 31, 2001 - 05:13 am:

I have a couple of problems. Can you help me, PLEASE!

(just show me how to set up the integrals)

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by

1) y=x2 , y=0 , x=-2 , x=-1 about y-axis

2) y=(x-1)1/2 , y=0 , x=5 about y=3 (I know the answer is "24 pi"; but I don't know how to set up.)

3) y=x2-3x+2; y=0 about y-axis.

I really appreciate your help!

By Jim Oldfield on Sunday, September 02, 2001 - 02:38 am:

Volume of revolution

When you have a curve as shown, you can rotate it around the y-axis to get a solid. To find the volume, divide the solid into thin slices (one is shown on the diagram) of width dx each. These slices are approximately cylinders of radius f(x), and as dx®0 the approximation becomes better and better. Therefore, if V is the volume of the solid and a & b are the start and end x values:
V=Sb ap[f(x)]2dx
=òa bp[f(x)]2dx
=...

For the second part, first subtract 3 from y (so y=...-3) then treat as if rotating about the y-axis.