## Volumes of Solids with Known Cross Sections

You can use the definite integral to find the volume of a solid with specific cross sections on an interval, provided you know a formula for the region determined by each cross section. If the cross sections generated are perpendicular to the *x*‐axis, then their areas will be functions of *x*, denoted by *A(x*). The volume ( *V*) of the solid on the interval [ *a, b*] is

If the cross sections are perpendicular to the *y*‐axis, then their areas will be functions of *y*, denoted by *A(y*). In this case, the volume ( *V*) of the solid on [ *a, b*] is

**Example 1:** Find the volume of the solid whose base is the region inside the circle *x* ^{2} + *y* ^{2} = 9 if cross sections taken perpendicular to the *y*‐axis are squares.

Because the cross sections are squares perpendicular to the *y*‐axis, the area of each cross section should be expressed as a function of *y*. The length of the side of the square is determined by two points on the circle *x* ^{2} + *y* ^{2} = 9 (Figure 1).

**Figure 1 **Diagram for Example 1.

The area ( *A*) of an arbitrary square cross section is *A* = *s* ^{2}, where

The volume ( *V*) of the solid is

**Example 2:** Find the volume of the solid whose base is the region bounded by the lines *x* + 4 *y* = 4, *x* = 0, and *y* = 0, if the cross sections taken perpendicular to the *x*‐axis are semicircles.

Because the cross sections are semicircles perpendicular to the *x*‐axis, the area of each cross section should be expressed as a function of *x*. The diameter of the semicircle is determined by a point on the line *x* + 4 *y* = 4 and a point on the *x*‐axis (Figure 2).

**Figure 2 **Diagram for Example 2.

The area ( *A*) of an arbitrary semicircle cross section is

The volume ( *V*) of the solid is