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Published byRegan Shelton Modified over 3 years ago

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We sometimes need an efficient method to estimate area when we can not find the antiderivative.

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Actual area under curve:

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Left-hand rectangular approximation: Approximate area: (too low)

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Approximate area: Right-hand rectangular approximation: (too high)

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Averaging the two: 1.25% error(too high)

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Averaging right and left rectangles gives us trapezoids:

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(still too high)

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Trapezoidal Rule: h = width of subinterval = (b – a)/n This gives us a better approximation than either left or right rectangles.

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Trapezoidal Rule: h = width of subinterval = (b – a)/n To see if the Trapezoidal Rule is an overestimate, underestimate, or exact, use the Concavity Test. If f’’(x) = 0, approximation is exact. If f’’(x) > 0, approximation is an overestimate If f’’(x) < 0, approximation is an underestimate.

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Example 1 Use the Trapezoidal Rule with n = 4 to estimate We must partition [1, 2] into four subintervals of equal length. x11.251.51.752 f(x)125/1636/1649/164

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Compare this with the Midpoint Rule: Approximate area: (too low)0.625% error The midpoint rule gives a closer approximation than the trapezoidal rule, but in the opposite direction.

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Midpoint Rule: (too low)0.625% error Trapezoidal Rule: 1.25% error (too high) Notice that the trapezoidal rule gives us an answer that has twice as much error as the midpoint rule, but in the opposite direction. If we use a weighted average: This is the exact answer!

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This weighted approximation gives us a closer approximation than the midpoint or trapezoidal rules. Midpoint: Trapezoidal: twice midpointtrapezoidal

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Simpson’s Rule: ( h = width of subinterval, n must be even ) Example:

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Simpson’s rule can also be interpreted as fitting parabolas to sections of the curve, which is why this example came out exactly. Simpson’s rule will usually give a very good approximation with relatively few subintervals. It is especially useful when we have no equation and the data points are determined experimentally.

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1 §12.4 The Definite Integral The student will learn about the area under a curve defining the definite integral.

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