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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the two rectangles is the
same as taking the area
of the trapezoid above
Trapezoidal rule approximates the integral
Use the Trapezoidal Rule to approximate
y = sin x
y = sin x
Snidly Fizbane Simpson
Use Simpson's Rule to approximate. Compare the results for
n = 4 and n = 8.
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
(Note: the 1st derivative test of
, which would be at x = 0.
The error is thus
Thus were one to pick n ≥ 3, one would have an error less than 0.01.
½ (SUM OF BASES) X height
THE FORMULA FOR THE AREA OF A TRAPAZOID!!!!!!
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: