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 ).
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Figure 1
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Diagram for Example 1.
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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 ).
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Figure 2
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Diagram for Example 2.
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The area (
A) of an arbitrary semicircle cross section is
The volume (
V) of the solid is
Review Topics
Applications of the Definite Integral
