Chapter 9 Techniques of Integration

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Chapter 9Techniques of Integration
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Chapter Outline

• Integration by Substitution
• Integration by Parts
• Evaluation of Definite Integrals
• Approximation of Definite Integrals
• Some Applications of the Integral
• Improper Integrals

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9.1
Integration by Substitution
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Section Outline

• Differentiation and Integration Formulas
• Integration by Substitution
• Using Integration by Substitution

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Differentiation Integration Formulas
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Integration by Substitution
If u g(x), then
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Using Integration by Substitution
EXAMPLE
Determine the integral by making an appropriate
substitution.
SOLUTION
Let u x2 2x 3, so that
. That is,
Therefore,
And so we have
Rewrite in terms of u.
Bring the factor 1/2 outside.
Integrate.
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Using Integration by Substitution
CONTINUED
Replace u with x2 2x 3.
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Using Integration by Substitution
EXAMPLE
Determine the integral by making an appropriate
substitution.
SOLUTION
Let u 1/x 2, so that
. Therefore,
And so we have
Rearrange factors.
Rewrite in terms of u.
Integrate.
Rewrite u as 1/x 2.
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Using Integration by Substitution
EXAMPLE
Determine the integral by making an appropriate
substitution.
SOLUTION
Let u cos2x, so that
. Therefore,
And so we have
Rearrange factors.
Rewrite in terms of u.
Integrate.
Rewrite u as cos2x.
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9.2
Integration by Parts
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Section Outline

• Integration by Parts
• Using Integration by Parts

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Integration by Parts
G(x) is an antiderivative of g(x).
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Using Integration by Parts
EXAMPLE
Evaluate.
SOLUTION
Our calculations can be set up as follows
Differentiate
Integrate
Then
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Using Integration by Parts
CONTINUED
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Using Integration by Parts
EXAMPLE
Evaluate.
SOLUTION
Our calculations can be set up as follows
Then
Notice that the resultant integral cannot yet be
solved using conventional methods. Therefore, we
will attempt to use integration by parts again.
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Using Integration by Parts
CONTINUED
Our calculations can be set up as follows
Then
Therefore, we have
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Using Integration by Parts
EXAMPLE
Evaluate.
SOLUTION
Our calculations can be set up as follows
Then
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Using Integration by Parts
CONTINUED
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9.3
Evaluation of Definite Integrals
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Section Outline

• The Definite Integral
• Evaluating Definite Integrals
• Change of Limits Rule
• Finding the Area Under a Curve
• Integration by Parts and Definite Integrals

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The Definite Integral
where F?(x) f (x).
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Evaluating Definite Integrals
EXAMPLE
Evaluate.
SOLUTION
First let u 1 2x and therefore du 2dx. So,
we have
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Evaluating Definite Integrals
CONTINUED
Consequently,
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Change of Limits Rule
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Using the Change of Limits Rule
EXAMPLE
Evaluate using the Change of Limits Rule.
SOLUTION
First let u 1 2x and therefore du 2dx.
When x 0 we have u 1 2(0) 1. And when x
1, u 1 2(1) 3. Thus
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Finding the Area Under a Curve
EXAMPLE
Find the area of the shaded region.
SOLUTION
To find the area of the shaded region, we will
integrate the given function. But we must know
what our limits of integration will be.
Therefore, we must determine the three
x-intercepts of the function.
This is the given function.
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Finding the Area Under a Curve
CONTINUED
Replace y with 0 to find the x-intercepts.
Set each factor equal to 0.
Solve for x.
Therefore, the left-most region (above the
x-axis) starts at x -3 and ends at x 0.
The right-most region (below the x-axis) starts
at x 0 and ends at x 3. So, to find the area
in the shaded regions, we will use the following.
Now lets find an antiderivative for both
integrals. We will use u 9 x2 and du -2xdx.
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Finding the Area Under a Curve
CONTINUED
Now we solve for the area.
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Finding the Area Under a Curve
CONTINUED
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Integration by Parts Definite Integrals
EXAMPLE
Evaluate.
SOLUTION
To solve this integral, we will need integration
by parts. Our calculations can be set up as
follows
Then
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Integration by Parts Definite Integrals
CONTINUED
Therefore, we have
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9.4
Approximation of Definite Integrals
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Section Outline

• The Midpoint Rule
• The Trapezoidal Rule
• Simpsons Rule
• Error Analysis

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The Midpoint Rule
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Using the Midpoint Rule
EXAMPLE
Approximate the following integral by the
midpoint rule.
SOLUTION
We have ?x (b a)/n (4 1)/3 1. The
endpoints of the four subintervals begin at a 1
and are spaced 1 unit apart. The first midpoint
is at a ?x/2 1.5. The midpoints are also
spaced 1 unit apart. According to the midpoint
rule, the integral is approximately equal to
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The Trapezoidal Rule
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Using the Trapezoidal Rule
EXAMPLE
Approximate the following integral by the
trapezoidal rule.
SOLUTION
As in the last example, ?x 1 and the endpoints
of the subintervals are a0 1, a1 2, a2 3,
and a3 4. The trapezoidal rule gives
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Simpsons Rule
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Using Simpsons Rule
EXAMPLE
Approximate the following integral by Simpsons
rule.
SOLUTION
As in the last example, ?x 1 and the endpoints
of the subintervals are a0 1, a1 2, a2 3,
and a3 4. Simpsons rule gives
NOTE Although this happens to be the exact
answer, remember that Simpsons Rule is still
just an approximation and therefore it generally
yields only an estimate of the exact answer, not
the exact answer itself.
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Error Analysis
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Using Error Analysis
EXAMPLE
Obtain a bound on the error of using the midpoint
rule with n 3 to approximate
SOLUTION
Here a 1, b 4, and f (x) (2x 3)3.
Differentiating twice, we find that
How large could f ??(x) be if x satisfies 1
x 4? Since the function 48x 72 is clearly
increasing on the interval from 1 to 4 (in fact,
its increasing everywhere), its greatest value
occurs at x 4. Therefore, its greatest value is
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Using Error Analysis
CONTINUED
so we may take A 120 in the preceding theorem.
The error of approximation using the midpoint
rule is at most
NOTE We have hitherto determined that the exact
value of this integral is 78 and that the
midpoint approximation for it (using n 3) is
72. Therefore, this approximation was in error
by 78 72 6. Our result in this exercise says
that our midpoint approximation error should be
no greater than 15. Since 6 is no greater than
15, this result suggests that our midpoint
approximation was done correctly.
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9.5
Some Applications of the Integral
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Section Outline

• The Riemann Sum
• Interest Compounded Continuously
• Continuous Stream of Income
• Population in a Ring

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The Riemann Sum
?x (b a)/n, t1, t2, ., tn are selected
points from a partition a, b.
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Interest Compounded Continuously
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Continuous Stream of Income
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Continuous Stream of Income
EXAMPLE
Find the present value of a stream of earnings
generated over the next 2 years at the rate of 50
7t thousand dollars per year at time t,
assuming a 10 interest rate.
SOLUTION
Using the theorem above, we have K(t) 50 7t,
r 0.06, T1 0 and T2 2. So, we have
To evaluate, we will need to rewrite the integral
as
and then evaluate the first integral directly and
the second using integration by parts. Upon
doing this, we have
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Continuous Stream of Income
CONTINUED
Now we evaluate the integral.
So the present value of the described stream of
earnings is 102.90.
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Population in a Ring
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Population in a Ring
EXAMPLE
The population density of Philadelphia in 1940
was given by the function 60e-0.4t.
Calculate the number of people who lived within
5 miles of the city center.
SOLUTION
Using the theorem above, we have D(t) 60e-0.4t,
a 0 and b 5. So, we have
Using integration by parts, we have
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9.6
Improper Integrals
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Section Outline

• Improper Integrals
• Evaluating Improper Integrals

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Improper Integrals
Definition Example
Improper Integral A tool used to calculate the area of a region that extends infinitely far to the right or left along the x-axis. Used when a limit of integration takes on a value of 8 or -8.
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Evaluating Improper Integrals
EXAMPLE
Evaluate the improper integral if it is
convergent.
SOLUTION
As b ? 8, approaches 8, so the
integral is divergent. Therefore, there is no
solution.
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Evaluating Improper Integrals
EXAMPLE
Evaluate the improper integral if it is
convergent.
SOLUTION
As b ? 8, approaches 0, so the integral
is convergent. Therefore,
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Evaluating Improper Integrals
EXAMPLE
Evaluate the improper integral if it is
convergent.
SOLUTION
As b ? -8, e4b approaches 0 so that
approaches 1/4. Thus the improper integral
converges and has value 1/4.