section 5.9 solutions

n the mass contributed by xi1 xi is approximately

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Unformatted text preview: , xn = b} be a partition of the interval [a, b]. Let xi∗ be a point in the ith interval, and let xi = xi − xi−1 , i = 1, 2, . . . , n. The mass contributed by [xi−1 , xi ] is approximately λ(xi∗ ) xi . The sum of these contributions, ∗ ∗ λ(x1 ) x1 + · · · + λ(xn ) xn , is a Riemann sum, which as ||P || → 0, gives the mass of the rod b M= λ(x) dx. (5. 9. 4) a Take Mi as the mass contributed by [xi−1 , xi ] . Then xMi Mi ∼ xi∗ λ(xi∗ ) = xi . Therefore xM M = xM1 M1 + · · · + xMn Mn ∼ x1 λ(x1 ) =∗ ∗ ∗ ∗ x1 + · · · + xn λ(xn ) xn . As ||P || → 0, the sum on the right converges...
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This document was uploaded on 11/11/2013.

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