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