Math 152 – Spring 2016
Section 7.2
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Section 7.2– Volume
What is the volume of a cylinder?
What if we have a shape where the cross-section changes?
Theorem. Let S be a solid that lies between the planes Pa and Pb . If the crosssectional area of S in the plane Px is A(x), where A is an integrable function, then the
volume of S is
Z
b
V =
A(x) dx
a
Example 1. Show that the volume of a sphere is V = 34 πr3 .
Math 152 – Spring 2016
Section 7.2
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Theorem. If the region bounded by the curves y = f (x), x = a, x = b, and the x-axis
is revolved around the x-axis, then the volume of the resulting solid is
b
Z
π[f (x)]2 dx
V =
a
Remark. This is called the disk method since the cross-sections we are integrating are
disks.
Example 2. Find the
√ volume of the solid obtained by rotating about the x-axis the
region bounded by y x − 1, x = 2, x = 5, and y = 0.
Remark. We can also rotate a region around the y-axis, but then we need to integrate
along the y-axis with y as the variable.
Theorem. If the region bounded by the curves x = g(y), y = c, y = d, and the y-axis
is revolved around the y-axis, then the volume of the resulting solid is
Z
V =
b
π[g(y)]2 dy
a
Example 3. Find the volume of the solid obtained by rotating about the y-axis the
region bounded by (y − 2)2 = x, x = 0, and y = 0.
Math 152 – Spring 2016
Section 7.2
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Example 4. Sketch the region bounded by the curves y = x2 and y = 4x. Find the
volume when this region is rotated around the x-axis.
Theorem. If the region bounded by y = f (x), y = g(x), x = a, and x = b is rotated
around the x-axis (with f (x) ≤ g(x) on [a, b]), then the resulting solid has volume
Z
V =π
b
[f (x)]2 − [g(x)]2 dx
a
Remark.
• This is called the washer method since the cross-sections look like
washers (a disk with a hole cut in the middle).
• If the rotation is not around the x- or y-axis, then you need to be careful to use
the correct outer and inner radius for the washer. The formula above only works
for rotating around the x- and y-axis.
Example
5. Find the volume of the solid obtained by rotating the region bounded by
√
y = x and y = x2 about the line x = 4.
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Section 7.2
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Example 6. Find the volume
√ of the solid obtained by rotating around the line y = 1
the region bounded by y = 3 x and y = x2 graphed below.
Non-Rotational Solids
Example 7. Find the volume of the following solids.
(a) The base of S is the triangular region with vertices (0, 0), (2, 0), and (0, 1). Crosssections perpendicular to the x-axis are semicircles.
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Section 7.2
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(b) The base of S is a circle with radius 2 and the cross-sections are equilateral triangles.
(c) The base of S is the area enclosed by x = 1 − y 2 and the y-axis. Cross-sections
perpendicular to the y-axis are isosceles triangles with height equal to the base.
Math 152 – Spring 2016
Section 7.2
(d) The base of S is the region bounded by y =
perpendicular to the x-axis are squares.
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√
x and y = x2 . Cross-sections
(e) Find the volume of a pyramid whose base is a square with side L and whose height
is h.