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7.5 Approximate Integration
http://classic.hippocampus.org/course_locator?course=General+Calculus+II&lesson=49&topic=4&width=800&height=684&topicTitle=Approximate+integration&skinPath=http%3A%2F%2Fclassic.hippocampus.org%2Fhippocampus.skins%2Fdefault
http://classic.hippocampus.org/course_locator?course=General+Calculus+II&lesson=49&topic=4&width=800&height=684&topicTitle=Approximate+integration&skinPath=http://classic.hippocampus.org/hippocampus.skins/defaulthttp://classic.hippocampus.org/course_locator?course=General+Calculus+II&lesson=49&topic=4&width=800&height=684&topicTitle=Approximate+integration&skinPath=http://classic.hippocampus.org/hippocampus.skins/defaulthttp://classic.hippocampus.org/course_locator?course=General+Calculus+II&lesson=49&topic=4&width=800&height=684&topicTitle=Approximate+integration&skinPath=http://classic.hippocampus.org/hippocampus.skins/defaulthttp://classic.hippocampus.org/course_locator?course=General+Calculus+II&lesson=49&topic=4&width=800&height=684&topicTitle=Approximate+integration&skinPath=http://classic.hippocampus.org/hippocampus.skins/defaulthttp://classic.hippocampus.org/course_locator?course=General+Calculus+II&lesson=49&topic=4&width=800&height=684&topicTitle=Approximate+integration&skinPath=http://classic.hippocampus.org/hippocampus.skins/defaulthttp://classic.hippocampus.org/course_locator?course=General+Calculus+II&lesson=49&topic=4&width=800&height=684&topicTitle=Approximate+integration&skinPath=http://classic.hippocampus.org/hippocampus.skins/default
3
There are two situations in which it is impossible to
find the exact value of a definite integral:
◦ When finding the antiderivative of a function is difficult or impossible
◦ When the function is determined by a scientific experiment through instrument readings or collected
data.
Ex:
21 1
3
0 1
1
xe dx or x dx
For those cases we will use Approximate Integration
We have used approximate integration on chapter 5
when we learned how to find areas under the curve.
There we approximated the area by using the Riemann
Sum by using the Left, Right and Midpoint rules.
Now we are going to learn two new methods:
The Trapezoid Rule and Simpson’s Rule
The Midpoint Rule states:
1 2
1 1
( ) ( ) ( ) ... ( )
1,
2
b
n n
a
i i i i i
f x dx M x f x f x f x
Where
b ax and x x x midpoint of x x
n
The Trapezoid rule approximates the integral by averaging the
approximations obtained by using the Left and Right Endpoint
Rules:
Trapezoid Rule
0 1 2( ) ( ) 2 ( ) 2 ( )... ( )2
b
n n
a
i
xf x dx T f x f x f x f x
b awhere x and x a i x
n
Use the midpoint and the trapezoid rule with n=5 to
approximate the integral
Solution
Example 1
2
1
1 dxx
https://mathfixation.files.wordpress.com/2012/06/section-71.pdf
Suppose
If ET and EM are the errors in the Trapezoid and the
Midpoint Rule respectively then
Errors Bounds in MP and Trap Rules
( ) f x K for a x b
3 3
2 212 24T M
K b a K b aE and E
n n
Find the Errors ET and EM in the previous example
Solution
Example 2
2
1
1 dxx
https://mathfixation.files.wordpress.com/2012/06/section-71.pdf
10
How large should we take n in order to guarantee
that the Trapezoid Rule and Midpoint Rule
approximations are accurate to within 0.0001 for
the integral below?
Solution
2
1
1 dxx
https://mathfixation.files.wordpress.com/2012/06/section-71.pdf
11
Simpson’s Rule uses parabolas to approximate integration
instead of straight line segments.
0 1 2 3
2 1
( ) ( ) 4 ( ) 2 ( ) 4 ( ) ...3
2 ( ) 4 ( ) ( )
b
n
a
n n n
xf x dx S f x f x f x f x
f x f x f x
b aWhere n is even and x
n
Suppose
If ES is the error involved in the Simpson’s Rule, then
Errors Bounds in Simpson’s Rule
4( ) f x K for a x b
5
4180S
K b aE
n
Example 4
13
Use the n=6 to approximate the given integral, rounding
to six decimal places by the
a) Midpoint Rule
b) Simpson’s Rule
Compare your answers to the actual value to determine the error in each approximation.
Solution
1
0
xe dx
https://mathfixation.files.wordpress.com/2012/06/section-71.pdf
Practice 1
14
Use the n=4 to approximate the given integral, rounding
to six decimal places by the
a) Trapezoid Rule
b) Midpoint Rule
c) Simpson’s Rule
Solution
12
2
0
sin x dx
https://mathfixation.files.wordpress.com/2012/06/section-71.pdf
Practice 2
15
Use the n=8 to approximate the given integral, rounding
to six decimal places by the
a) Trapezoid Rule
b) Midpoint Rule
c) Simpson’s Rule
Solution
4
0
1 x dx
https://mathfixation.files.wordpress.com/2012/06/section-71.pdf
Practice 3
16
Use the n=10 to approximate the given integral, rounding
to six decimal places by the
a) Trapezoid Rule
b) Midpoint Rule
c) Simpson’s Rule
6
3
4
ln 2x dx
Practice 4
17
a) Find the approximations T10 and M10 for the integral
below
b) Estimate the errors in approximation from above
c) How large do we have to choose n so that the
approximations Tn and Mn are accurate to within 0.0001
21
1
xe dx
Practice 5
18
a) The table below gives the power consumption P in
megawatts in San Diego County from midnight to 6:00
AM on December 8, 1999. Use Simpson’s Rule to
estimate the energy used during that time period. (Use
the fact that power is the derivative of energy)
Solution
t P t P
0:00 1814 3:30 1611
0:30 1735 4:00 1621
1:00 1686 4:30 1666
1:30 1646 5:00 1745
2:00 1637 5:30 1886
2:30 1609 6:00 2052
3:00 1604
https://mathfixation.files.wordpress.com/2012/06/section-71.pdf
http://youtu.be/JGeCLfLaKMw
http://youtu.be/z_AdoS-ab2w
http://www4.ncsu.edu/~acherto/NCSU/MA241/sec59.pdf
http://youtu.be/zUEuKrxgHws
http://youtu.be/z_AdoS-ab2whttp://youtu.be/z_AdoS-ab2whttp://youtu.be/z_AdoS-ab2whttp://youtu.be/z_AdoS-ab2whttp://youtu.be/z_AdoS-ab2whttp://youtu.be/z_AdoS-ab2whttp://www4.ncsu.edu/~acherto/NCSU/MA241/sec59.pdfhttp://www4.ncsu.edu/~acherto/NCSU/MA241/sec59.pdfhttp://youtu.be/zUEuKrxgHws