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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- May 30th 2006, 01:51 PMSusie38Calculus Problem
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. - May 30th 2006, 11:43 PMTD!
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}

$