Title: Volume

Definition 1

Let  be continuous and nonnegative on [a, b]. Then, the volume of the solid generated by revolving the area below the graph of  about the x-axis between  and  is defined by

Example 1:

Find the volume of the solid generated by revolving the region bounded by the graph of  and the axis  about the x-axis.

Solution:

 and the x-axis intersect at  and  since

Thus,

Definition 2:

If the area is revolved about the y-axis, then the volume of the solid generated is defined by

Example 2:

Find the volume of the solid generated by revolving the area below the graph of  and the x-axis about the y-axis.

Solution:

 and the x-axis intersect at  and  since

Then,

Volume by rotating the area between two curves

Let . The volume generated by rotating the region bounded by the curves and lines  and  around the x-axis is defined by

Example 3:

Find the volume of the solid generated by revolving the area bounded by the graphs of  and  about the x-axis.

Solution:

The two lines intersect at the points  and  since

Thus,

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