Answer to Question #223877 in Chemical Engineering for Lokika

Question

Evaluate ∫y=0^{1}∫x=y^{2√2-y^{2}}y/2√x^{2}+y^{2}dxdy ,by clange the order of integration.

Answer 1

$∫_{y=0}^{1}∫_{x=y}^{2\sqrt{2-y^{2}}} \frac{y}{2}\sqrt{x^{2}+y^{2}}dxdy\\ ∫_{x=0}^{1}∫_{y=x}^{\sqrt{2-\frac{x^2}{4}}} \frac{y}{2}\sqrt{x^{2}+y^{2}}dydx\\ =\int _0^1\left(\frac{\left(3x^2+8\right)^{\frac{3}{2}}}{48}-\frac{\sqrt{2}x^3}{3}\right)dx\\ =\frac{-48\sqrt{3}\ln \left(2\right)+32\sqrt{3}\ln \left(\sqrt{3}+\sqrt{11}\right)+23\sqrt{11}}{192}-\frac{1}{6\sqrt{2}}\\ =0.44671$