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$f(x,y) = \sqrt{2} - \sqrt{x^2 + y}$ $g(x) = \sqrt{1 - x^2}$ $h(y) = \pm \frac{\sqrt{-y \pm2\sqrt{y}+1}}{\sqrt{2}}$ Given $f(x,y)$, $g(x)$, $h(y)$, an infinitesimal quantity $\Delta$, and $\{x \in \mathbb{R}, y \in \mathbb{R}\}$ how do I evaluate and simplify the following: $$ \int_{0}^{h(y + \Delta)}{\int_{0}^{x}{f(x,y+\Delta) dx} dy} - \int_{0}^{h(y)}{\int_{0}^{x}{f(x,y) dx} dy} + \int_{0}^{\pm h(1)}{\int_{0}^{\pm1}{f(x,y) dx} dy} = \int_{0}^{x} g(x)$$ for $y$ approaching zero in the range $[0,1]$ (and let the integral evaluate to zero for all $y$ outside this range)? The result should be a function of one variable.
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Fri, 18 Jun 2021 18:27 GMT