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Use S 2 to estimate 1 2 1 x d x .

25 36

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Key concepts

  • We can use numerical integration to estimate the values of definite integrals when a closed form of the integral is difficult to find or when an approximate value only of the definite integral is needed.
  • The most commonly used techniques for numerical integration are the midpoint rule, trapezoidal rule, and Simpson’s rule.
  • The midpoint rule approximates the definite integral using rectangular regions whereas the trapezoidal rule approximates the definite integral using trapezoidal approximations.
  • Simpson’s rule approximates the definite integral by first approximating the original function using piecewise quadratic functions.

Key equations

  • Midpoint rule
    M n = i = 1 n f ( m i ) Δ x
  • Trapezoidal rule
    T n = 1 2 Δ x ( f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + + 2 f ( x n 1 ) + f ( x n ) )
  • Simpson’s rule
    S n = Δ x 3 ( f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + 2 f ( x 4 ) + 4 f ( x 5 ) + + 2 f ( x n 2 ) + 4 f ( x n 1 ) + f ( x n ) )
  • Error bound for midpoint rule
    Error in M n M ( b a ) 3 24 n 2
  • Error bound for trapezoidal rule
    Error in T n M ( b a ) 3 12 n 2
  • Error bound for Simpson’s rule
    Error in S n M ( b a ) 5 180 n 4

Approximate the following integrals using either the midpoint rule, trapezoidal rule, or Simpson’s rule as indicated. (Round answers to three decimal places.)

1 2 d x x ; trapezoidal rule; n = 5

0.696

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0 3 4 + x 3 d x ; trapezoidal rule; n = 6

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0 3 4 + x 3 d x ; Simpson’s rule; n = 3

9.279

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0 12 x 2 d x ; midpoint rule; n = 6

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0 1 sin 2 ( π x ) d x ; midpoint rule; n = 3

0.5000

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Use the midpoint rule with eight subdivisions to estimate 2 4 x 2 d x .

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Use the trapezoidal rule with four subdivisions to estimate 2 4 x 2 d x .

T 4 = 18.75

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Find the exact value of 2 4 x 2 d x . Find the error of approximation between the exact value and the value calculated using the trapezoidal rule with four subdivisions. Draw a graph to illustrate.

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Approximate the integral to three decimal places using the indicated rule.

0 1 sin 2 ( π x ) d x ; trapezoidal rule; n = 6

0.500

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0 3 1 1 + x 3 d x ; trapezoidal rule; n = 6

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0 3 1 1 + x 3 d x ; Simpson’s rule; n = 3

1.1614

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0 0.8 e x 2 d x ; trapezoidal rule; n = 4

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0 0.8 e x 2 d x ; Simpson’s rule; n = 4

0.6577

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0 0.4 sin ( x 2 ) d x ; trapezoidal rule; n = 4

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0 0.4 sin ( x 2 ) d x ; Simpson’s rule; n = 4

0.0213

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0.1 0.5 cos x x d x ; trapezoidal rule; n = 4

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0.1 0.5 cos x x d x ; Simpson’s rule; n = 4

1.5629

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Evaluate 0 1 d x 1 + x 2 exactly and show that the result is π / 4 . Then, find the approximate value of the integral using the trapezoidal rule with n = 4 subdivisions. Use the result to approximate the value of π .

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Approximate 2 4 1 ln x d x using the midpoint rule with four subdivisions to four decimal places.

1.9133

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Approximate 2 4 1 ln x d x using the trapezoidal rule with eight subdivisions to four decimal places.

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Use the trapezoidal rule with four subdivisions to estimate 0 0.8 x 3 d x to four decimal places.

T(4) = 0.1088

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Use the trapezoidal rule with four subdivisions to estimate 0 0.8 x 3 d x . Compare this value with the exact value and find the error estimate.

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Using Simpson’s rule with four subdivisions, find 0 π / 2 cos ( x ) d x .

1.0

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Show that the exact value of 0 1 x e x d x = 1 2 e . Find the absolute error if you approximate the integral using the midpoint rule with 16 subdivisions.

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Given 0 1 x e x d x = 1 2 e , use the trapezoidal rule with 16 subdivisions to approximate the integral and find the absolute error.

Approximate error is 0.000325.

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Practice Key Terms 5

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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