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« Area Between Two Curves
Applications Of The Definite Integrals: (lesson 2 of 3)

Volume

Definition 1

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

Volume formula

Volume

Example 1:

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

Solution:

volume solution and the x-axis intersect at and volume solution since

volume solution

Thus,

limit

Definition 2:

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

volume definition

volume definition

Example 2:

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

Solution:

volume example and the x-axis intersect at volume solution and volume solution since

volume

Then,

volume example

Volume by rotating the area between two curves

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

volume formula

Example 3:

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

Solution:

The two lines intersect at the points volume solution and volume solution since

Thus,

volume solution