Numerical Integration

AP Calculus

Numerical Integration 

* Used when normal definite integration is not possible. 

a). When there is no elementary function for the anti-derivative; 

i.e.: or  

b). Data is given in tabular or graphical form and it is too much effort to find the representative function.

cosx xdx31 x dx

I. RIEMANN’S SUMS

REM: Riemann’s Sum uses Rectangles to approximate the accumulation.

A = bh => h - Left Endpoints

- Right Endpoints - Midpoint

The more accurate is the Midpoint Sum (must remember how to use all three – Left, Right, and Midpoint). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Midpoint Rule: with

 

In Words: The width of the subinterval times the sum of the heights AT THE MIDPOINT of each subinterval.

1

( ) ( )nb

i ian i

f x dx f c xLim

1

1

( )2

nbi i

ai

x xf x dx f x

b a

xn

0 1 11 2 ...2 2 2

n nx x x xx xb aA f f f

n

Illustration:

Function:  on [ 0 , 4 ] with n = 4

 

 

2( ) 4f x x x 4 2

0(4 )x x dx

Example: Graphical

Find the Average Revenue for the 5 years.

II. TRAPEZOID METHOD:

 Uses Trapezoids to fill the regions rather than rectangles:

REM:     

  

---------------------------------------------------------------

1 2

1( )

2A h b b

0 1

1( )( ( ) ( ))

2

b aA f x f x

n

1 0 1

1( ( ) ( ))

2

b aA f x f x

n

2

1

2

b aA

n

1 2( ( ) ( ))f x f x

3

1

2

b aA

n

2 3( ( ) ( ))f x f x

0 1 2 3

1( ( ) 2 ( ) 2 ( ) ( ))

2

b aA f x f x f x f x

n

(Notice this is the average of the Left and Right Riemann's Sums)

Trapezoid Rule:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

 In Words: One half * width of subinterval * the ( 1 , 2 , 2 , … , 2 , 1 ) pattern of the heights found at the points of the subinterval.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

1( ) ( ) ( )

2

nb

i iai

b af x dx f x f x

n

0 1 2 1

1( ( ) 2 ( ) 2 ( ) ... 2 ( ) ( ))

2 n n

b af x f x f x f x f x

n

Illustration: (Trapezoid)

Function:  on [ 0 , 4 ] with n = 4

 

 

2( ) 4f x x x 4 2

0(4 )x x dx

The data for the acceleration a(t) of a car from 1 to 15 seconds are given in the table below. If the velocity at t = 0 is 5 ft/sec, which of the following gives the approximate velocity at t = 15 using the Trapezoidal Rule?

t (sec) 0 3 6 9 12 15

a(t) (ft/sec2)

4 8 6 9 10 10

Example: Data

A lot is bounded by a stream. and two straight roads that meet at right angles. Use the Trapezoid Rule to approximate the area of the lot (x and y are measured in square meters)

III. SIMPSON’S METHOD

 

Built on: The area of the region below a quadratic function.

REM: Three points are required to write a quadratic equation

since the equation has 3 variables; A,B,C in Therefore, to get the 3 points needed, Simpson’s uses double subintervals to approximate the accumulation.. . . . . . . . . . . . . . . . . . . . . . . . . . . .

THEOREM:

with n even

2( )b

aAx Bx C dx

2y Ax Bx C

21 2 3

1( ) ( ( ) 4 ( ) ( ))

3

b

a

b aAx Bx C dx f x f x f x

n

Simpson’s: (cont)

EXTENDED:  

__________________________________________________ 

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simpson’s Formula:

 

Note the pattern: 1,4,2,4,1 1,4,2,4,2,4,1 etc

1 2 3

1( ( ) 4 ( ) ( ))

3

b af x f x f x

n

1

3

b a

n

3 4 5( ( ) 4 ( ) ( ))f x f x f x

1 2 3 4 5

1[ ( ) 4 ( ) 2 ( ) 4 ( ) ( )]

3

b af x f x f x f x f x

n

1

2

1

( ) ( )i

i

xnb

ai x

f x dx Ax Bx C dx

1 2 3 4 5

1[ ( ) 4 ( ) 2 ( ) 4 ( ) ( )]

3

b af x f x f x f x f x

n

Illustration: (Simpson’s)

Function:  on [ 0 , 4 ] with n = 4

 

 

2( ) 4f x x x 4 2

0(4 )x x dx

Although the economy is continuously changing, we analyze it with discrete measurements. The following table records the annual inflation rate as measured each month for 13 consecutive months. Use Simpson’s Rule with n = 12 to find the overall inflation rate for the year.

Month Annual Rate

January 0.04

February 0.04

March 0.05

April 0.06

May 0.05

June 0.04

July 0.04

August 0.05

September 0.04

October 0.06

November 0.06

December 0.05

January 0.05

Example: Graphical - all three

Error:

Approximation gives rise to two questions>>>>

1) How close are we to the actual answer? and

2) How do we get close enough?

Error: MIDPOINT

MnE Error using Midpoint with n partitions

3

2

( )( )

24Mn

b aE f c

n

C is an un-findable number in [a,b] but whose existence is guaranteed; therefore do “ERROR BOUND”

3

2

( )

24Mn i

b aE M

n

Where Mi the MAX of

on i[a,b] ( )f x

Error: TRAPEZIOD

TnE Error using Trapezoid with n partitions

3

2

( )( )

12Tn

b aE f c

n

C is an un-findable number in [a,b] but whose existence is guaranteed; therefore do “ERROR BOUND”

3

2

( )

12Tn i

b aE M

n

Where Mi the MAX of

on i[a,b] ( )f x

Example: How close are we?

Approximate using Trapezoid Method

with 4 intervals and find the Error bound.

12

0

(1 )x dx

Example: How many intervals are required?

Approximate using Midpoint Rule

to within .

14

0

5x dx11000

Error: SIMPSONS

SnE Error using Simpson’s with n partitions

5

4

( )( )

180S IVn

b aE f c

n

C is an un-findable number in [a,b] but whose existence is guaranteed; therefore do “ERROR BOUND”

5

4

( )

180Sn i

b aE M

n

Where Mi the MAX of

on i[a,b] ( )IVf x

Last Update:

• 02/05/10