section 5.9 solutions

B 37 f x dx a x1 f x dx x0 x2 f x dx

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Unformatted text preview: that, if the averages are the same on every interval, then the functions are everywhere the same. b 37. f (x) dx = a x1 f (x) dx + x0 x2 f (x) dx + · · · + x1 xn f (x) dx xn−1 By the mean-value theorem for integrals, there exists a number xi∗ ∈ (xxi−1 , xi ) such that xi xi−1 f (x) dx = f (xi∗ ) (xi − xi−1 ) = f (xi∗ ) xi , i = 1, 2, . . . , n 39. (a) A rod is lying on the x-axis from x = a to x = b. The mass density of the rod is given by the continuous function λ = λ(x). Let P = {a = x0 , x1 , x2 , . . . , xn−1...
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