Notes #15-Trapezoidal Rule and Simpson Rule PDF

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Professor Bianca Santoro...

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Notes #15-The Trapezoidal Rule: 1

Some integrals are very difficult to integrate like

e

x2

dx . So we need to approximate.

0

The Trapezoidal Rule is a way to approximate the definite integral. This is useful because all functions do NOT have antiderivatives, and all shapes do not have functions to represent them. The Trapezoidal Rule, which uses area of trapezoids, is a much better approximation than rectangles. 

The trapezoidal Rule uses the area of a trapezoid which is A=



The trapezoidal Rule states:

b

f (x )dx  a



b a [ f (x 0 )  2 f (x 1 )  2 f (x 2 )  2 f (x 3 )  ...f (x n )] 2n

1 h(b1  b2 ) 2

This rule must be memorized.

The trapezoidal rule approximates the region between the graph of f and the x-axis by using “n” trapezoids of equal widths. b



f (x )dx  the sum of the areas of the trapezoids. a 2

1. Use the trapezoidal rule to find

1

xdx

using n=5 (ans .6956)

1

1

2. Use the trapezoidal Rule with 4 equal subdivisions to approximate

x

4 dx 1

ans (2.8125)

1

3. Use the trapezoidal Rule with 4 trapezoids to evaluate the following

e

x2

dx

(ans 1.462)

0

2

4. Use the trapezoidal Rule with 4 trapezoids to approximate

3

x dx 0

3

5. Use the trapezoidal Rule for

1

16  x

0

2

dx use n=6 (ans .160675)

(ans 17/4)



6. Use the Trapezoidal Rule with n=4 for

sin xdx 0

7

8

(ans

 2 2 2 8





Simpson’s Rule: Simpson's Rule is another way to APPROXIMATE the definite integral. Simpson's Rule, which uses area of parabolas, is sometimes a better approximation than the Trapezoidal rule. It depends on the shape of the curve. Simpson's rule only works if n is an EVEN number. Simpson's Rule:

NOTE: For Simpson's Rule, the pattern of the coefficients is: 1 4 2 4 2 4 2 4 2 ... 4 2 4 1

Example of Simpson’s Rule:

to approximate

with n = 6

Solution: From the integral above we can read the following.

1.

(a)Trapezoidal Rule

(b)Simpson’s Rule...