Explanation Given information: ∫ − 1 8 2 x d x Calculation: We have ∫ − 1 8 2 x d x . The left Riemann sum used the left endpoints of a subinterval: ∫ a b f ( x ) d x ≈ Δ x ( f ( x 0 ) + f ( x 1 ) + 2 f ( x 2 ) + … + f ( x n − 2 ) + f ( x n − 1 ) ) , where Δ x = b − a n . So, Δ x = 8 + 1 3 = 3 Divide the interval [−1, 8] into n = 3 subintervals of length ?x = 2 a = [−1, 2], [0, 5], [5, 8] = b. Now, we just evaluate the function at the left endpoints: f ( x 0 ) = f ( a ) = f ( − 1 ) = 0.5 f ( x 1 ) = f ( 2 ) = 4 f ( x 2 ) = f ( 5 ) = 32 Finally, just sum up the above values and multiply by ?x = 3. ∫ a b f ( x ) d x ≈ Δ x ( f ( x 0 ) + f ( x 1 ) + 2 f ( x 2 ) + … + f ( x n − 2 ) + f ( x n − 1 ) ) ≈ 3 ( 0

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Chapter 5.2, Problem 9P

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Estimate the integeral ∫−182xdx using a left-hand and a right-hand sum with n = 3.

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Chapter 5.2, Problem 8P

Chapter 5.2, Problem 10P

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Estimate the integeral ∫ − 1 8 2 x d x using a left-hand and a right-hand sum with n = 3.

Solution Summary: The author explains how to estimate the integeral displaystyle and the right-hand sum with n = 3. The left Riemann sum uses the left endpoints of a subinterval.

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Estimate the integeral ∫−182xdx using a left-hand and a right-hand sum with n = 3.

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Chapter 5 Solutions