Explanation Given: The provided integral is: ∫ 3 4 1 x − 2 d x And the value n = 4 . Formula used: 1. According to trapezoidal rule ∫ a b f ( x ) d x ≈ b − a 2 n [ f ( x 0 ) + 2 f ( x 1 ) + .... + 2 f ( x n − 1 ) + f ( x n ) ] 2. According to Simpson’s rule ∫ a b p ( x ) d x ≈ ( b − a 3 n ) [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + ...2 f ( x n − 2 ) + 4 f ( x n − 1 ) + f ( x n ) ] 3. The log rule for integration : ∫ d x x = log | x | + C Calculation: Consider the following function f ( x ) = 1 x − 2 . From the given integral we get, b = 4 , a = 3 and n = 4 . Now, calculate the width as: Δ x = b − a n = 4 − 3 4 = 1 4 Use trapezoidal rule to approximate the value of integral, ∫ 3 4 1 x − 2 d x ≈ 4 − 3 2 ( 4 ) [ f ( 3 ) + 2 f ( 3 + 1 4 ) + 2 f ( 3 + 1 4 × 2 ) + 2 f ( 3 + 1 4 × 3 ) + f ( 3 + 1 4 × 4 ) ] = 1 8 [ f ( 3 ) + 2 f ( 13 4 ) + 2 f ( 7 2 ) + 2 f ( 15 4 ) + f ( 4 ) ] = 1 8 [ ( 1 3 − 2 ) + 2 ( 1 ( 13 / 4 ) − 2 ) + 2 ( 1 ( 7 / 2 ) − 2 ) + 2 ( 1 ( 15 / 4 ) − 2 ) + ( 1 4 − 2 ) ] = 1 8 [ 1 + 2 ( 4 5 ) + 2 ( 2 3 ) + 2 ( 4 7 ) + 1 2 ] By solving further, we get, 1 8 [ 1 + 2 ( 4 5 ) + 2 ( 2 3 ) + 2 ( 4 7 ) + 1 2 ] = 1 8 [ 1 + 8 5 + 4 3 + 8 7 + 1 2 ] = 1 8 [ 1 + 1

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Chapter 8.6, Problem 5E

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To calculate: The approximate value of definite integral ∫341x−2dx for given value of n by using Trapezoidal rule and Simpson’s rule and compare it with the exact value of the definite integral.

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Chapter 8.6, Problem 4E

Chapter 8.6, Problem 6E

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The approximate value of definite integral ∫ 3 4 1 x − 2 d x for given value of n by using Trapezoidal rule and Simpson’s rule and compare it with the exact value of the definite integral.

Solution Summary: The author compares the approximate value of the definite integral displaystyle and Simpson's rule with the exact value.

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To calculate: The approximate value of definite integral ∫341x−2dx for given value of n by using Trapezoidal rule and Simpson’s rule and compare it with the exact value of the definite integral.

Want to see the full answer?

Chapter 8 Solutions