Question Number 115364 by Bird last updated on 25/Sep/20
![calculate ∫∫_([0,1]^2 ) (√(xy))(x^2 +y^2 )dxdy](https://www.tinkutara.com/question/Q115364.png)
$${calculate}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:\:\sqrt{{xy}}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right){dxdy} \\ $$
Answered by Olaf last updated on 25/Sep/20
![∫∫_([0,1]^2 ) ((√y)x^(5/2) +y^(5/2) (√x))dxdy = ∫_0 ^1 [(2/7)(√y)x^(7/2) +(2/3)y^(5/2) x^(3/2) ]_0 ^1 dy = ∫_0 ^1 ((2/7)(√y)+(2/3)y^(5/2) )dy = [(2/7)×(2/3)y^(3/2) +(2/3)×(2/7)y^(7/2) ]_0 ^1 = (8/(21))](https://www.tinkutara.com/question/Q115431.png)
$$\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \left(\sqrt{{y}}{x}^{\mathrm{5}/\mathrm{2}} +{y}^{\mathrm{5}/\mathrm{2}} \sqrt{{x}}\right){dxdy} \\ $$$$=\:\int_{\mathrm{0}} ^{\mathrm{1}} \left[\frac{\mathrm{2}}{\mathrm{7}}\sqrt{{y}}{x}^{\mathrm{7}/\mathrm{2}} +\frac{\mathrm{2}}{\mathrm{3}}{y}^{\mathrm{5}/\mathrm{2}} {x}^{\mathrm{3}/\mathrm{2}} \right]_{\mathrm{0}} ^{\mathrm{1}} {dy} \\ $$$$=\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\frac{\mathrm{2}}{\mathrm{7}}\sqrt{{y}}+\frac{\mathrm{2}}{\mathrm{3}}{y}^{\mathrm{5}/\mathrm{2}} \right){dy} \\ $$$$=\:\left[\frac{\mathrm{2}}{\mathrm{7}}×\frac{\mathrm{2}}{\mathrm{3}}{y}^{\mathrm{3}/\mathrm{2}} +\frac{\mathrm{2}}{\mathrm{3}}×\frac{\mathrm{2}}{\mathrm{7}}{y}^{\mathrm{7}/\mathrm{2}} \right]_{\mathrm{0}} ^{\mathrm{1}} \\ $$$$=\:\frac{\mathrm{8}}{\mathrm{21}} \\ $$
Commented by mathmax by abdo last updated on 25/Sep/20
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir} \\ $$