Calc05_5 - 5.5 Numerical Integration Mt Shasta California...

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5.5 Numerical Integration Mt. Shasta, California Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 1998
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Using integrals to find area works extremely well as long as we can find the antiderivative of the function. Sometimes, the function is too complicated to find the antiderivative. At other times, we don’t even have a function, but only measurements taken from real life. What we need is an efficient method to estimate area when we can not find the antiderivative.
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2 1 1 8 y x = + 4 3 0 1 24 A x x = + 4 2 0 1 1 8 A x dx = + 0 4 x Actual area under curve: 20 3 A = 6.6 =
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2 1 1 8 y x = + 0 4 x Left-hand rectangular approximation: Approximate area: 1 1 1 3 1 1 1 2 5 5.75 8 2 8 4 + + + = = (too low)
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Approximate area: 1 1 1 3 1 1 2 3 7 7.75 8 2 8 4 + + + = = 2 1 1 8 y x = + 0 4 x Right-hand rectangular approximation: (too high)
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Averaging the two: 7.75 5.75 6.75 2 + = 1.25% error (too high)
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Averaging right and left rectangles gives us trapezoids: 1 9 1 9 3 1 3 17 1 17 1 3 2 8 2 8 2 2 2 8 2 8 T = + + + + + + +
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