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« Area Between Two Curves |

Applications Of The Definite Integrals: (lesson 2 of 3)

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,

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,

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,