Explanation Use the given integral to determine the actual integral value and check if that is the same as the value of S 2 and hence verify if the given statement is true. Given: ∫ a b x 3 d x Calculation: Given ∫ a b x 3 d x Let f ( x ) = x 3 . Also, Δ x = ( b − a ) / 2 and the endpoints are a , ( a + b ) / 2 , b With this, we get S 2 ( x 3 ) = 1 3 ( b − a 2 ) [ a 3 + 4 ( a + b 2 ) 3 + b 3 ] = ( b − a 6 ) ( 3 2 ) [ a 3 + a 2 b + a b 2 + b 3 ] = > S 2 ( x 3 ) = ( b − a 4 ) [ a 3 + a 2 b + a b 2 + b 3 ] = 1 4 ( b 4 − a 4 ) Also, by integration we get ∫ a b x 3 d x = 1 4 ( b 4 − a 4 ) By using the result from the previous exercise, we have ∫ a b ( s x 3 + p x 2 + q x + r ) d x = s ∫ a b x 3 d x + ∫ a b ( p x 2 + q x

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 8.8, Problem 64E

To determine

To show that $S_{2}$ also gives the exact value for $\int_{a}^{b} x^{3} d x$ and conclude that $S_{N}$ is exact for all cubic polynomials and show by counter-example that $S_{2}$ is not exact for integrals of $x^{4}$

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Chapter 8.8, Problem 63E

Chapter 8.8, Problem 65E

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Chapter 8 Solutions

CALCULUS W/SAPLING ACCESS >IC<

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To show that S 2 also gives the exact value for ∫ a b x 3 d x and conclude that S N is exact for all cubic polynomials and show by counter-example that S 2 is not exact for integrals of x 4

Solution Summary: The author demonstrates that S 2 gives the exact value for a b x 3 d and concludes S N is exact for all cubic polynomials.

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To show that $S_{2}$ also gives the exact value for $\int_{a}^{b} x^{3} d x$ and conclude that $S_{N}$ is exact for all cubic polynomials and show by counter-example that $S_{2}$ is not exact for integrals of $x^{4}$

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To show that S 2 also gives the exact value for ∫ a b x 3 d x and conclude that S N is exact for all cubic polynomials and show by counter-example that S 2 is not exact for integrals of x 4

Solution Summary: The author demonstrates that S 2 gives the exact value for a b x 3 d and concludes S N is exact for all cubic polynomials.

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To show that S2 also gives the exact value for ∫abx3dx and conclude that SNis exact for all cubic polynomials and show by counter-example that S2is not exact for integrals of x4

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Chapter 8 Solutions

To show that $S_{2}$ also gives the exact value for $\int_{a}^{b} x^{3} d x$ and conclude that $S_{N}$ is exact for all cubic polynomials and show by counter-example that $S_{2}$ is not exact for integrals of $x^{4}$