Question Number 226366 by klipto last updated on 26/Nov/25
$$\boldsymbol{\mathrm{compute}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{double}}\:\boldsymbol{\mathrm{integral}} \\ $$$$\int_{\boldsymbol{\mathrm{y}}=\mathrm{0}} ^{\mathrm{1}} \int_{\boldsymbol{\mathrm{x}}=\mathrm{0}} ^{\mathrm{2}} \boldsymbol{\mathrm{x}}^{\mathrm{2}} \boldsymbol{\mathrm{dxdy}}\:\boldsymbol{\mathrm{and}}\:\:\int_{\boldsymbol{\mathrm{y}}=\mathrm{0}} ^{\mathrm{1}} \int_{\boldsymbol{\mathrm{x}}=\mathrm{0}} ^{\mathrm{2}} \boldsymbol{\mathrm{y}}^{\mathrm{2}} \boldsymbol{\mathrm{dxdy}} \\ $$$$ \\ $$
Answered by fantastic2 last updated on 26/Nov/25
![1) ∫_(y=0) ^1 (∫_(x=0) ^2 x^2 dx)dy =∫_(y=0) ^1 ([(x^3 /3)]_0 ^2 )dy =∫_0 ^1 (8/3)dy=(8/3)[y]_0 ^1 =(8/3) 2) ∫_(y=0) ^1 (y^2 ∫_(x=0) ^2 dx)dy =∫_(y=0) ^1 2y^2 dy =2[(y^3 /3)]_0 ^1 =(2/3)](https://www.tinkutara.com/question/Q226370.png)
$$\left.\mathrm{1}\right) \\ $$$$\int_{{y}=\mathrm{0}} ^{\mathrm{1}} \left(\int_{{x}=\mathrm{0}} ^{\mathrm{2}} {x}^{\mathrm{2}} {dx}\right){dy} \\ $$$$=\int_{{y}=\mathrm{0}} ^{\mathrm{1}} \left(\left[\frac{{x}^{\mathrm{3}} }{\mathrm{3}}\right]_{\mathrm{0}} ^{\mathrm{2}} \right){dy} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{8}}{\mathrm{3}}{dy}=\frac{\mathrm{8}}{\mathrm{3}}\left[{y}\right]_{\mathrm{0}} ^{\mathrm{1}} =\frac{\mathrm{8}}{\mathrm{3}} \\ $$$$\left.\mathrm{2}\right) \\ $$$$\int_{{y}=\mathrm{0}} ^{\mathrm{1}} \left({y}^{\mathrm{2}} \int_{{x}=\mathrm{0}} ^{\mathrm{2}} {dx}\right){dy} \\ $$$$=\int_{{y}=\mathrm{0}} ^{\mathrm{1}} \mathrm{2}{y}^{\mathrm{2}} {dy} \\ $$$$=\mathrm{2}\left[\frac{{y}^{\mathrm{3}} }{\mathrm{3}}\right]_{\mathrm{0}} ^{\mathrm{1}} =\frac{\mathrm{2}}{\mathrm{3}} \\ $$