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Half the battle here is having an adequate picture of the volume in question. In all the methods here you cut the region into slices in some way, compute the volume of each slice, and integrate over them.

To do this successfully you must make sure that you are considering the right figure (rotated about the right axis), and you must know the volumes of your slices and endpoints. The rest is routine.

There is a quick way to get the same answer for the volume  of an ellipsoid of revolution as considered here.

This figure has a radius 1 in the y and z direction, but radius 2 in the x direction. If we take a sphere of radius 1 (and volume 4p / 3) and blow it up in the x direction by a factor of two, we get just such a figure.

We can deduce that the volume is .

If all three of its radi are different, r1, r2 and r3, its volume is  since it can be made into a sphere of radius r1 by appropriate scale changes of and .

These scale changes each change the volume by the same factor.