We use cross-sectional area to compute volume.

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### Examples

The base of a solid region is bounded by the curves , , and . The cross sections
perpendicular to the axis are squares. Compute the volume of the region.

- Solution:
- Lines in the -plane which are perpendicular to the -axis are vertical, so the base of a typical cross section will extend from to . Since each cross section will have area To compute volume, we integrate between and , since these are the most extreme values of found in our region. Therefore

The base of a solid region is bounded by the curves , , and . The cross sections
perpendicular to the -axis are squares. Compute the volume of the region.

- Solution:
- Lines in the -plane which are perpendicular to the -axis are horizontalvertical , so the base of a typical cross section will extend from the graph to the graph . The length of the base is the difference of -coordinates (since all points on a slice have the same -coordinate), so the length of the base is , giving the square an area of (note that the answer is a function of because different cross sections will generally have different areas). To compute volume, we integrate between and , since these are the most extreme values of found in our region (note that we can find the upper value by solving for the intersection of the curves and ). Therefore we integrate to conclude