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CHAPTER 4 SECTION 4.6 NUMERICAL INTEGRATION. Theorem 4.16 The Trapezoidal Rule. Trapezoidal Rule. a. b. Which dimension is the h? Which is the b 1 and the b 2. b 2. h. b 1. •. Instead of calculating approximation rectangles we will use trapezoids More accuracy Area of a trapezoid.

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trapezoidal rule
Trapezoidal Rule

a

b

  • Which dimension is the h?
  • Which is the b1 and the b2

b2

h

b1

  • Instead of calculatingapproximation rectangleswe will use trapezoids
    • More accuracy
  • Area of a trapezoid
slide5

Averaging the areas of

the two rectangles is the

same as taking the area

of the trapezoid above

the subinterval.

trapezoidal rule6
Trapezoidal Rule

f(xi-1)

f(xi)

dx

Trapezoidal rule approximates the integral

approximating with the trapezoidal rule
Approximating with the Trapezoidal Rule

Use the Trapezoidal Rule to approximate

y = sin x

1

Four subintervals

y = sin x

1

Eight subintervals

simpson s rule
Simpson's Rule

a

b

Snidly Fizbane Simpson

  • As before, we dividethe interval into n parts
    • n must be even
  • Instead of straight lines wedraw parabolas through each group of three consecutive points
    • This approximates the original curve for finding definite integral – formula shown below
approximate using simpson s rule
Approximate using Simpson's Rule.

Use Simpson's Rule to approximate. Compare the results for

n = 4 and n = 8.

slide15

The Approximate Error in the Trapezoidal Rule:

Determine a value n such that the Trapezoidal Rule will approximate the value ofwith an error less than 0.01.

First find the second derivative of

slide16

The maximum of occurs at x = 0 (see the graph below)

(Note: the 1st derivative test of

gives

, which would be at x = 0.

slide17

And at x = 0, f '' = 1

The error is thus

Thus were one to pick n ≥ 3, one would have an error less than 0.01.

slide19

NOTE: THEY SIMPLY USED

½ (SUM OF BASES) X height

THE FORMULA FOR THE AREA OF A TRAPAZOID!!!!!!

slide20
John is stressed out. He's got all kinds of projects going on, and they're all falling due around the same time. He's trying to stay calm, but despite all of his yoga techniques, relaxation methods, and sudden switch to decaffeinated coffee, his nerves are still on edge. Forget the 50-page paper due in two weeks and the 90-minute oral presentation next Monday morning; he still hasn't done laundry in weeks, and his stink is beginning to turn heads. Because of all the stress (and possibly due to his lack of bathing), John's started to lose his hair.
  • Below is a chart representing John's rate of hair loss (in follicles per day) on various days throughout a two-week period. Use 6 trapezoids to approximate John's total hair loss over that traumatic 14-day period.
slide21

You may be tempted to use the Trapezoid Rule, but you can't use that handy formula, because not all of the trapezoids have the same width. Between day 1 and 4, for example, the width of the approximating trapezoid will be 3, but the next trapezoid will be 6 – 4 = 2 units wide. Therefore, you need to calculate each trapezoid's area separately, knowing that the area of a trapezoid is equal to one-half of the product of the width of the trapezoid and the sum of the bases: