Math 1B
§ 6.2 Volumes
Overview: The volume of a solid can be computed by “integrating its cross sectional area”.
Think of slicing the solid with a knife through x and computing the area of this slice.
b
Volume =
∫ A( x )dx
a
Here’s the reason: Approximate the volume by slicing it into thin cylindrical “slabs”.
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Adding the volumes of these slabs, we get an approximation for the total volume.
n
Volume ≈ ∑ A( x *i )Δx
i=1
The approximation gets better as the slabs get thinner (as n → ∞ ), so
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n
Volume = lim ∑ A( x *i )Δx
n →∞
i=1
But we recognize the right side as a definite integral. Therefore,
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b
Volume =
∫ A( x )dx
a
Stewart – 7e
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Solids of Revolution
Let f be nonnegative and continuous on [ a,b] and let R be the region bounded above by the graph of
f , below by the x-axis, and on the sides by x = a and x = b .
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y = f(x)
R
a
b
When this region is revolved about the x-axis, it
generates a solid having circular cross sections.
Since the cross section at x has radius
f ( x ) , its area is A ( x ) =
Volume =
Since the cross sections are disk shaped, this application is called the method of disks.
Example: Find the volume of the solid obtained when the region under the curve y = x 2 from 0 to 1 is
revolved about the x-axis.
Stewart – 7e
2
Example: Find the volume of the solid obtained when the region enclosed by y =
1
, y = 1 , y = 2 , and
x
the y-axis is revolved about the y-axis.
Stewart – 7e
3
Suppose f and g are nonnegative continuous functions on [ a,b] such that g( x ) ≤ f ( x ) . Let R be the
region bounded by the graphs of f and g, x = a , and x = b .
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R
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When this region is revolved about the xaxis, it generates a solid having washer
shaped cross sections.
Since the cross section at x has inner radius g ( x ) and outer radius f ( x ) , its area is
A ( x) =
Volume =
This application is called the method of washers.
Stewart – 7e
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Example: Revolve the region between the graphs of y = x 2 and y = x3 about the x-axis. Find the
volume of the resulting solid.
Stewart – 7e
5
Example: Find the volume of the solid obtained by rotating the region in the previous example about
the y-axis.
Stewart – 7e
6
Example: Now rotate the same region about the line y = 1 .
Stewart – 7e
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Example: Now rotate the same region about the line x = 2 .
Stewart – 7e
8