Integration Questions

Question Number 98672 by  M±th+et+s last updated on 15/Jun/20

$$\int_{\mathrm{0}} ^{\mathrm{4}} \int_{\mathrm{0}} ^{\frac{{x}}{\mathrm{4}}} {e}^{{x}^{\mathrm{2}} } \:{dx}\:{dy} \\$$

Answered by Ar Brandon last updated on 16/Jun/20

$$\mathrm{Let}\:\mathcal{I}=\int_{\mathrm{0}} ^{\frac{\mathrm{x}}{\mathrm{4}}} \mathrm{e}^{\mathrm{x}^{\mathrm{2}} } \mathrm{dx}........\left(\mathrm{1}\right) \\$$ $$\Rightarrow\:\:\:\mathcal{I}=\int_{\mathrm{0}} ^{\frac{\mathrm{x}}{\mathrm{4}}} \mathrm{e}^{\mathrm{y}^{\mathrm{2}} } \mathrm{dy}........\left(\mathrm{2}\right) \\$$ $$\Rightarrow\mathcal{I}^{\mathrm{2}} =\int_{\mathrm{0}} ^{\frac{\mathrm{x}}{\mathrm{4}}} \int_{\mathrm{0}} ^{\frac{\mathrm{x}}{\mathrm{4}}} \mathrm{e}^{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} } \mathrm{dxdy}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \int_{\mathrm{0}} ^{\frac{\mathrm{x}}{\mathrm{4}}} \mathrm{re}^{\mathrm{r}^{\mathrm{2}} } \mathrm{drd}\theta \\$$ $$\Rightarrow\mathcal{I}^{\mathrm{2}} =\left[\frac{\theta}{\mathrm{2}}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left[\mathrm{e}_{} ^{\mathrm{r}^{\mathrm{2}^{} } } \right]_{\mathrm{0}} ^{\frac{\mathrm{x}}{\mathrm{4}}} =\frac{\pi}{\mathrm{4}}\left[\:\mathrm{e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{16}}} −\mathrm{1}_{} \right] \\$$ $$\Rightarrow\mathcal{I}=\pm\frac{\sqrt{\pi}}{\mathrm{2}}\left[\:\mathrm{e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{16}}} −\mathrm{1}_{} \right]^{\frac{\mathrm{1}}{\mathrm{2}}} \\$$ $$\Rightarrow\int_{\mathrm{0}} ^{\mathrm{4}} \int_{\mathrm{0}} ^{\frac{{x}}{\mathrm{4}}} {e}^{{x}^{\mathrm{2}} } \:{dx}\:{dy}=\pm\mathrm{2}\sqrt{\pi}\left[\:\mathrm{e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{16}}} −\mathrm{1}_{} \right]^{\frac{\mathrm{1}}{\mathrm{2}}} \\$$

Commented by M±th+et+s last updated on 16/Jun/20

$${nice}\:{work}\:{and}\:{correct}\:{thank}\:{you} \\$$

Commented byAr Brandon last updated on 16/Jun/20

I don't know if I have done it in the required manner. Can someone kindly check ?

Commented byAr Brandon last updated on 16/Jun/20