Integration Questions

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

$$\int_{\mathrm{0}} ^{\mathrm{3}} \int_{\mathrm{0}} ^{\mathrm{2}} \mathrm{x}^{\mathrm{2}} \mathrm{ydydx}=\int_{\mathrm{0}} ^{\mathrm{3}} \mathrm{x}^{\mathrm{2}} \mathrm{dx}\int_{\mathrm{0}} ^{\mathrm{2}} \mathrm{ydy}=\left[\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{3}}\right]_{\mathrm{0}} ^{\mathrm{3}} ×\left[\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{2}}\right]_{\mathrm{0}} ^{\mathrm{2}} \\$$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{9}×\mathrm{2}=\mathrm{18} \\$$

Answered by gungun last updated on 16/Apr/23

$$\int_{\mathrm{0}} ^{\mathrm{3}} \left(\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} {y}^{\mathrm{2}} \right){dx} \\$$ $$=\int_{\mathrm{0}} ^{\mathrm{3}} \left(\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} {y}^{\mathrm{2}} \right]_{\mathrm{0}} ^{\mathrm{2}} {dx} \\$$ $$=\int_{\mathrm{0}} ^{\mathrm{3}} \left(\mathrm{2}{x}^{\mathrm{2}} \right)\:{dx} \\$$ $$=\left(\frac{\mathrm{2}}{\mathrm{3}}{x}^{\mathrm{3}} \right]_{\mathrm{0}} ^{\mathrm{3}} =\frac{\mathrm{2}}{\mathrm{3}}×\mathrm{3}^{\mathrm{3}} =\mathrm{9}×\mathrm{2}=\mathrm{18} \\$$ $$\: \\$$