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Volumes of Revolution

Rotation About the x-axis

Integration can be used to find the area of a region bounded by a curve whose equation you know. If we want to find the area under the curve y = x2 between x = 0 and x = 5, for example, we simply integrate x2 with limits 0 and 5.
Now imagine that a curve, for example y = x2, is rotated around the x-axis so that a solid is formed. The volume of the shape that is formed can be found using the formula:

Rotation about the y-axis

If the body is rotated about the y-axis rather than the x-axis, then we use the formula: