7.6 Numerical Integration

7.6 Numerical Integration - parabola uniquely How many...

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1 7.6 Numerical Integration In applications, it is sometimes necessary to calculate an integral of a func- tion, for which no explicit antiderivative can be obtained. For example, when the function is given to us as a list of samples over a prescribed time interval (say daily temperatures for a month). In this section we study two methods of numerical integration: Trapezoidal Rule and Simpson’s Rule . Trapezoidal Rule a b x y f H x L Area of i th trapezoid = Total area of all trapezoids = Definition Trapezoidal Rule To approximate Z b a f ( x ) dx use: T = where y i = f ( x i ), Δ x = b - a n and n is a number of subintervals REMARK: Trapezoidal rule can be derived as an average of the left- and right-endpoint Riemann sums, i.e.: T = S L + S R 2
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2 Example 1 Approximate Z 2 1 sin ( e x ) dx , n = 6 x f ( x ) T = Example 2 Approximate Z 4 1 x dx , n = 3 x f ( x ) T =
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3 Simpson’s Rule Simpson’s rule calculates the area under a parabola that approximates the graph of the function. a b x y f H x L How many points we need to describe a
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Unformatted text preview: parabola uniquely? How many subintervals do these three points create? As a result, in order to apply Simp-son’s rule the number of subintervals ( n ) MUST be . x x 1 x 2 x y 4 Definition Simpson’s Rule To approximate Z b a f ( x ) dx use: S Simp = where y i = f ( x i ), Δ x = b-a n and n is an even number of subintervals Example 3 Approximate Z 2 1 sin ( e x ) dx , n = 6 x f ( x ) S Simp = Example 4 Approximate Z 4 1 √ x dx , n = 4 x f ( x ) S Simp = 5 Example 5 To do Using the following table of values approximate Z-3 f ( x ) dx x-3-2.5-2-1.5-1-0.5 f ( x ) 2 1 1 1 2 3 • Using trapezoidal rule, T = • Using Simpson’s rule, S Simp = Example 6 To do Set up the Simpson’s rule approximation for the integral Z 9 3 x x dx , n = 4. Do not evaluate...
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This note was uploaded on 10/02/2011 for the course MATH 1206 taught by Professor Llhanks during the Fall '08 term at Virginia Tech.

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7.6 Numerical Integration - parabola uniquely How many...

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