LEt a and b be positive integers and y = f(x) be a decreasing differentiable function sucha that f(0) = b and f(a) = 0. Prove that the integral of 2xy dx from 0 to a equals the integral of x^2 dy from 0 to b.
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Well, 2xdx = d(x²), the notice the symmetry: $\displaystyle \int\limits_0^a {2xydx} = \int\limits_0^b {x^2 dy} \Leftrightarrow \int\limits_0^a {ydx^2 } = \int\limits_0^b {x^2 dy} $
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