Answer to Question #224863 in Chemical Engineering for Lokika

Question #224863

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


1
Expert's answer
2021-08-16T02:32:54-0400

y=01x=y22y2y2x2+y2dxdy∫_{y=0}^{1}∫_{x=y}^{2 \sqrt{2-y^{2}}}\frac{y}{2\sqrt{x^{2}+y^{2}}}dxdy\\

In this x and y are from 0<y<1 and y<x<22y20<y<1 \space and \space y<x< 2\sqrt{2-y^2}\\

So we can change it as 0<x<1 and x<y<2x240<x<1 \space and \space x<y< \sqrt{2-\frac{x^2}{4}}\\

y=01x=y22y2y2x2+y2dxdy=x=01y=x2x24y2x2+y2dydx01x2x24y2x2+y2dydx=14(3x2+822x)=0114(3x2+822x)dx=14(112+23ln(7+334)2)=14(112+23ln(7+334)2)=0.39554∫_{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\int _x^{\sqrt{2-\frac{x^2}{4}}}\frac{y}{2\sqrt{x^2+y^2}}dydx\\ =\frac{1}{4}\left(\sqrt{3x^2+8}-2\sqrt{2}x\right)\\ =\int _0^1\frac{1}{4}\left(\sqrt{3x^2+8}-2\sqrt{2}x\right)dx\\ =\frac{1}{4}\left(\frac{\sqrt{11}}{2}+\frac{2}{\sqrt{3}}\ln \left(\frac{7+\sqrt{33}}{4}\right)-\sqrt{2}\right)\\ =\frac{1}{4}\left(\frac{\sqrt{11}}{2}+\frac{2}{\sqrt{3}}\ln \left(\frac{7+\sqrt{33}}{4}\right)-\sqrt{2}\right)\\ =0.39554


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