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Chapter 7 – Techniques of Integration

7.7 Approximation Integration

7.7 Approximation Integration Erickson

7.7 Approximation Integration2

Why do we use Approximate Integration? 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

If the function is determined from a scientific experiment through instrument readings or collected data. There may not be a formula for the function.

1

1

31

0

x1or 2

dxdxe x

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7.7 Approximation Integration3

Approximate Integration

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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 using the Riemann Sums. We used Left, Right and Midpoint rules.

Now we are going to learn two new methods:

The Trapezoid Rule and Simpson’s Rule

Let’s compare the approximation methods.

7.7 Approximation Integration4

Midpoint Rule Remember, the midpoint rule states that

where

and

1 2( ) ...b

n n

a

f x dx M x f x f x f x

n

abx

iiiii xxxxx , ofmidpoint 2

111

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7.7 Approximation Integration5

Trapezoidal Rule The Trapezoid rule approximates the integral by

averaging the approximations obtained by using the Left and Right Endpoint Rules:

where

and

0 1 1( ) 2 ... 22

b

n n n

a

xf x dx T f x f x f x f x

n

abx

xiaxi

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7.7 Approximation Integration6

Example 1 Use (a) the Midpoint Rule and (b) the Trapezoidal Rule

with n = 5 to approximate the integral below. Round your answer to six decimal places.

2

1

1dx

x

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7.7 Approximation Integration7

Error Bounds in MP and Trap Rules

Suppose for a ≤ x ≤ b.

If ET and EM are the errors in the Trapezoidal and

Midpoint Rules, then

Kxf |)("|

2

3

2

3

and ( )

12

( )

4

2MT

K b aE

K b aE

n n

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7.7 Approximation Integration8

Example 2 Find the error in the previous problem.

Previous problem:

Use (a) the Midpoint Rule and (b) the Trapezoidal Rule with n = 5 to approximate the integral

2

1

1dx

x

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7.7 Approximation Integration9

Example 3 How large should we take n in order to guarantee that the

Trapezoidal Rule and Midpoint Rule approximations are accurate to within 0.0001 for the integral below?

2

1

1dx

x

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7.7 Approximation Integration10

Simpson’s Rule Simpson’s Rule uses parabolas to approximate

integration instead of straight line segments.

where

and n is even.

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

n

abx

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7.7 Approximation Integration11

Error Bounds in Simpson’s Rule

Suppose for a ≤ x ≤ b.

If ES is the error involved using Simpson’s Rule, then

Kxf |)(| )4(

5

4

( )

180S

K b aE

n

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7.7 Approximation Integration12

Example 4 Use the (a) Midpoint Rule and (b) Simpson’s Rule to

approximate the given integral with the specified value of n. Round your answers to six decimal places. Compare your results to the actual value to determine the error in each approximation.

1

0

6xe dx n

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7.7 Approximation Integration13

Example 5 Use (a) the Trapezoidal Rule,

(b) the Midpoint Rule, and

(c) Simpson’s Rule to approximate the given integral with the specified value of n. Round your answers to six decimal places.

1/2

2

0

sin , 4x dx n

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7.7 Approximation Integration14

Example 6 Use (a) the Trapezoidal Rule,

(b) the Midpoint Rule, and

(c) Simpson’s Rule to approximate the given integral with the specified value of n. Round your answers to six decimal places.

8,14

0

ndxx

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7.7 Approximation Integration15

Example 7 Use (a) the Trapezoidal Rule,

(b) the Midpoint Rule, and

(c) Simpson’s Rule to approximate the given integral with the specified value of n. Round your answers to six decimal places.

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6

3

4

ln 2 , 10x dx n

7.7 Approximation Integration16

Example 8

(a) Find the approximations T10 and M10 for the above integral.

(b) Estimate the errors in approximation of part (a). (c) How large do we have to choose n so that the

approximations Tn and Mn to the integral part (a) are accurate to within 0.0001?

dxe x2

1

/1

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7.7 Approximation Integration17

Example 9 The table (supplied by San Diego Gas and Electric) 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.)

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

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7.7 Approximation Integration18

Book Resources

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Video Examples Example 2 – pg. 510 Example 3 – pg. 511 Example 5 – pg. 513

More Videos Using the Midpoint Rule to approximate definite integrals , Part I Using the Midpoint Rule to approximate definite integrals, Part 2 Using the Midpoint Rule to approximate definite integrals, Part 3 The Trapezoidal Rule Using the Trapezoidal Rule to approximate an integral Errors in the Trapezoidal Rule and Simpson’s Rule Simpson’s Rule Using Simpson’s Rule to Approximate an Integral

7.7 Approximation Integration19

Book Resources

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Wolfram Demonstrations Comparing Basic Numerical Integration Methods

7.7 Approximation Integration20

Web Links

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http://youtu.be/JGeCLfLaKMw

http://youtu.be/z_AdoS-ab2w

http://www4.ncsu.edu/~acherto/NCSU/MA241/sec59.pdf

http://youtu.be/zUEuKrxgHws