Volumes Associated with Integrals in Spherical Coordinates

Example: Find the volume asssociated with

Solution: As in necessary for the expression of a volume, the within the integral is not important in assessing the volume associated with the triple integral. Although it obviously is for its evaluation.

Working from the outside inward, the first datum from the integrals is indicating that our volume will contain values of between and

The second datum from the integral is indicating that for every value of between and , we will accept values of that reside between and .

The third datum from the integral is indicating that for every value of and in the defined region, we will accept values of that reside between and . Taking all of the data into account, this triple integral will represent the following volume.