Question Number 190984 by Spillover last updated on 15/Apr/23
$$\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{3}} \int_{\mathrm{0}} ^{\mathrm{2}} \mathrm{x}^{\mathrm{2}} \mathrm{ydydx}\: \\ $$$$ \\ $$
Answered by a.lgnaoui last updated on 15/Apr/23
![∫_0 ^3 ∫_0 ^2 x^2 ydydx=∫_0 ^3 x^2 dx∫_0 ^2 ydy=[(x^3 /3)]_0 ^3 ×[(y^2 /2)]_0 ^2 =9×2=18](https://www.tinkutara.com/question/Q190986.png)
$$\int_{\mathrm{0}} ^{\mathrm{3}} \int_{\mathrm{0}} ^{\mathrm{2}} \mathrm{x}^{\mathrm{2}} \mathrm{ydydx}=\int_{\mathrm{0}} ^{\mathrm{3}} \mathrm{x}^{\mathrm{2}} \mathrm{dx}\int_{\mathrm{0}} ^{\mathrm{2}} \mathrm{ydy}=\left[\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{3}}\right]_{\mathrm{0}} ^{\mathrm{3}} ×\left[\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{2}}\right]_{\mathrm{0}} ^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{9}×\mathrm{2}=\mathrm{18} \\ $$
Answered by gungun last updated on 16/Apr/23
![∫_0 ^3 ((1/2)x^2 y^2 )dx =∫_0 ^3 ((1/2)x^2 y^2 ]_0 ^2 dx =∫_0 ^3 (2x^2 ) dx =((2/3)x^3 ]_0 ^3 =(2/3)×3^3 =9×2=18](https://www.tinkutara.com/question/Q191006.png)
$$\int_{\mathrm{0}} ^{\mathrm{3}} \left(\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} {y}^{\mathrm{2}} \right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{3}} \left(\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} {y}^{\mathrm{2}} \right]_{\mathrm{0}} ^{\mathrm{2}} {dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{3}} \left(\mathrm{2}{x}^{\mathrm{2}} \right)\:{dx} \\ $$$$=\left(\frac{\mathrm{2}}{\mathrm{3}}{x}^{\mathrm{3}} \right]_{\mathrm{0}} ^{\mathrm{3}} =\frac{\mathrm{2}}{\mathrm{3}}×\mathrm{3}^{\mathrm{3}} =\mathrm{9}×\mathrm{2}=\mathrm{18} \\ $$$$\: \\ $$