application of integration
TRANSCRIPT
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APPICATION OF INTEGRATIONAPPICATION OF INTEGRATION
SARDAR VALLABHBHAI PATEL INSTITAUTE OF TECHNOLOGY
GTU
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CONTENTCONTENT
1.Volume by cylindrical shell2.Volume of solids of revolutions by
washers method
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APPLICATIONS OF INTEGRATIONAPPLICATIONS OF INTEGRATION
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Volumes by Cylindrical Shells
APPLICATIONS OF INTEGRATION
In this section, we will learn:How to apply the method of cylindrical shells
to find out the volume of a solid.
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Let’s consider the problem of finding thevolume of the solid obtained by rotating aboutthe y-axis the region bounded by y = 2x2 - x3
and y = 0.
VOLUMES BY CYLINDRICAL SHELLS
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If we slice perpendicular to the y-axis, we get a washer.
However, to compute the inner radius and the outer radius of the washer, we would have to solve the cubic equation y = 2x2 - x3 for x in terms of y.
That’s not easy.
VOLUMES BY CYLINDRICAL SHELLS
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Fortunately, there is a method—the method of cylindrical shells—that is easier to use in such a case.
VOLUMES BY CYLINDRICAL SHELLS
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The figure shows a cylindrical shell with inner radius r1, outer radius r2,
and height h.
CYLINDRICAL SHELLS METHOD
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Its volume V is calculated by subtractingthe volume V1 of the inner cylinder from
the volume of the outer cylinder V2 .
CYLINDRICAL SHELLS METHOD
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Thus, we have:
2 1
2 22 1
2 22 1
2 1 2 1
2 12 1
( )( )( )
2 ( )2
V V V
r h r h
r r hr r r r hr r
h r r
CYLINDRICAL SHELLS METHOD
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Let ∆r = r2 – r1 (thickness of the shell) and
(average radius of the shell).
Then, this formula for the volume of a cylindrical shell becomes:
2V rh r
Formula 1
12 12r r r
CYLINDRICAL SHELLS METHOD
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The equation can be remembered as:
V = [circumference] [height] [thickness]
CYLINDRICAL SHELLS METHOD
2V rh r
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Now, let S be the solid obtained by rotating about the y-axis the region bounded by y = f(x) [where f(x) ≥ 0], y = 0, x = a and x = b, where b > a ≥ 0.
CYLINDRICAL SHELLS METHOD
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Divide the interval [a, b] into n subintervals [xi - 1, xi ] of equal width and let be
the midpoint of the i th subinterval.
xix
CYLINDRICAL SHELLS METHOD
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The rectangle with base [xi - 1, xi ] and height is rotated about the y-axis.
The result is a cylindrical shell with average radius , height , and thickness ∆x.
( )if x
( )if xix
CYLINDRICAL SHELLS METHOD
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Thus, by Formula 1, its volume iscalculated as follows:
(2 )[ ( )]i i iV x f x x
CYLINDRICAL SHELLS METHOD
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So, an approximation to the volume V of S is given by the sum of the volumes of these shells:
1 1
2 ( )n n
i i ii i
V V x f x x
CYLINDRICAL SHELLS METHOD
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The approximation appears to become betteras n →∞.
However, from the definition of an integral, we know that:
1
lim 2 ( ) 2 ( )n b
i i an i
x f x x x f x dx
CYLINDRICAL SHELLS METHOD
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Thus, the following appears plausible.
The volume of the solid obtained by rotating about the y-axis the region under the curve y = f(x) from a to b, is:
where 0 ≤ a < b
2 ( )b
aV xf x dx
Formula 2CYLINDRICAL SHELLS METHOD
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The argument using cylindrical shellsmakes Formula 2 seem reasonable, but later we will be able to prove it.
CYLINDRICAL SHELLS METHOD
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Here’s the best way to remember the formula.
Think of a typical shell, cut and flattened, with radius x, circumference 2πx, height f(x), and thickness ∆x or dx:
2 ( )b
athicknesscircumference height
x f x dx
CYLINDRICAL SHELLS METHOD
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This type of reasoning will be helpful in other situations—such as when we rotate about lines other than the y-axis.
CYLINDRICAL SHELLS METHOD
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Find the volume of the solid obtained byrotating about the y-axis the regionbounded by y = 2x2 - x3 and y = 0.
Example 1CYLINDRICAL SHELLS METHOD
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We see that a typical shell has radius x, circumference 2πx, and height f(x) = 2x2 - x3.
Example 1CYLINDRICAL SHELLS METHOD
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So, by the shell method, the volume is:
2 2 3
0
2 3 4
0
24 51 12 5 0
32 165 5
2 2
2 (2 )
2
2 8
V x x x dx
x x x dx
x x
Example 1CYLINDRICAL SHELLS METHOD
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It can be verified that the shell methodgives the same answer as slicing.
The figure shows a computer-generated picture of the solid whose volume we computed in the example.
Example 1CYLINDRICAL SHELLS METHOD
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Comparing the solution of Example 1 with the remarks at the beginning of the section, we see that the cylindrical shells method is much easier than the washer method for the problem.
We did not have to find the coordinates of the local maximum.
We did not have to solve the equation of the curve for x in terms of y.
NOTE
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However, in other examples, the methods learned in Section 6.2 may be easier.
NOTE
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Find the volume of the solid obtained by rotating about the y-axis the regionbetween y = x and y = x2.
Example 2CYLINDRICAL SHELLS METHOD
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The region and a typical shell are shown here.
We see that the shell has radius x, circumference 2πx, and height x - x2.
Example 2CYLINDRICAL SHELLS METHOD
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Thus, the volume of the solid is:
1 2
0
1 2 3
0
13 4
0
2
2
23 4 6
V x x x dx
x x dx
x x
Example 2CYLINDRICAL SHELLS METHOD
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As the following example shows, the shell method works just as well if we rotate about the x-axis.
We simply have to draw a diagram to identify the radius and height of a shell.
CYLINDRICAL SHELLS METHOD
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Use cylindrical shells to find the volume of the solid obtained by rotating about the x-axis the region under the curve from 0 to 1.
This problem was solved using disks in Example 2 in Section 6.2
y x
Example 3CYLINDRICAL SHELLS METHOD
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To use shells, we relabel the curve
as x = y2.
For rotation about the x-axis, we see that a typical shell has radius y, circumference 2πy, and height 1 - y2.
y x
Example 3CYLINDRICAL SHELLS METHOD
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So, the volume is:
In this problem, the disk method was simpler.
1 2
0
1 3
0
12 4
0
2 1
2 ( )
22 4 2
V y y dy
y y dy
y y
Example 3CYLINDRICAL SHELLS METHOD
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Find the volume of the solid obtained by rotating the region bounded by y = x - x2
and y = 0 about the line x = 2.
Example 4CYLINDRICAL SHELLS METHOD
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The figures show the region and a cylindricalshell formed by rotation about the line x = 2, which has radius 2 - x, circumference 2π(2 - x), and height x - x2.
Example 4CYLINDRICAL SHELLS METHOD
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So, the volume of the solid is:
0 2
1
0 3 2
1
143 2
0
2 2
2 3 2
24 2
V x x x dx
x x x dx
x x x
Example 4CYLINDRICAL SHELLS METHOD
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Volumes of Solids of Revolution: Washer Method
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Washers!
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Washers
Consider the area between two functions rotated about the axis f(x)
a b
g(x)
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Washers
Consider the area between two functions rotated about the axis
Now we have a hollow solid
f(x)
a b
g(x)
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Washers
Consider the area between two functions rotated about the axis
Now we have a hollow solidWe will sum the volumes of washers
f(x)
a b
g(x)
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Washers
f(x)
a b
g(x) 2 2( ) ( )
b
a
V f x g x dx
Outer Function
InnerFunction
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Find the volume of the solid formed by revolving the region bounded by y = (x) and y = x² over the interval [0, 1] about the x – axis.
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Find the volume of the solid formed by revolving the region bounded by y = (x) and y = x² over the interval [0, 1] about the x – axis. 2 2([ ( )] [ ( )] )
b
a
V f x g x dx
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Find the volume of the solid formed by revolving the region bounded by y = (x) and y = x² over the interval [0, 1] about the x – axis. 2 2([ ( )] [ ( )] )
b
a
V f x g x dx
1
0
222dxxxV
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Find the volume of the solid formed by revolving the region bounded by y = (x) and y = x² over the interval [0, 1] about the x – axis. 2 2([ ( )] [ ( )] )
b
a
V f x g x dx
1
0
222dxxxV
1
0
4 dxxxV
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Find the volume of the solid formed by revolving the region bounded by y = (x) and y = x² over the interval [0, 1] about the x – axis. 2 2([ ( )] [ ( )] )
b
a
V f x g x dx
1
0
222dxxxV
1
0
4 dxxxV 1
0
52
51
21
xxV
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Find the volume of the solid formed by revolving the region bounded by y = (x) and y = x² over the interval [0, 1] about the x – axis. 2 2([ ( )] [ ( )] )
b
a
V f x g x dx
1
0
222dxxxV
1
0
4 dxxxV 1
0
52
51
21
xxV
103