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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 x2 + y2 = 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 x2 + y2 = 9 (Figure 1 ).





Figure 1

Diagram for Example 1.


The area ( A) of an arbitrary square cross section is A = s2, 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




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