Integration Questions

Question Number 53292 by gunawan last updated on 20/Jan/19

$$\int_{\mathrm{0}} ^{\mathrm{1}} {e}^{{x}^{\mathrm{2}} } {dx}=.. \\$$

Commented bymaxmathsup by imad last updated on 20/Jan/19

$${we}\:{have}\:{e}^{{x}^{\mathrm{2}} } =\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{2}{n}} }{{n}!}\:\Rightarrow\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{{x}^{\mathrm{2}} } {dx}\:=\sum_{{n}=\:\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{{n}!}\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{2}{n}} {dx} \\$$ $$=\sum_{{n}=\mathrm{0}} ^{\infty\:} \:\:\:\:\:\frac{\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right){n}!}\:=\mathrm{1}\:+\frac{\mathrm{1}}{\mathrm{3}}\:+\frac{\mathrm{1}}{\mathrm{5}.\left(\mathrm{2}!\right)}\:+\frac{\mathrm{1}}{\mathrm{7}\left(\mathrm{3}!\right)}\:+\frac{\mathrm{1}}{\mathrm{9}\left(\mathrm{4}!\right)}\:+.... \\$$ $${and}\:{we}\:{get}\:{a}\:{best}\:{approximation}\:{if}\:{we}\:{take}\:\mathrm{10}\:{terms}\:{of}\:{this}\:{serie}. \\$$

Answered by tanmay.chaudhury50@gmail.com last updated on 20/Jan/19

$${aprroximation} \\$$ $${f}\left({x}\right)={e}^{{x}^{\mathrm{2}} } \:\:{f}\left(\mathrm{0}\right)=\mathrm{1}\:{f}\left(\mathrm{1}\right)={e} \\$$ $${trapazium}\:{area}=\frac{\mathrm{1}}{\mathrm{2}}\left[{f}\left(\mathrm{0}\right)+{f}\left(\mathrm{1}\right)\right]×\left(\mathrm{1}−\mathrm{0}\right) \\$$ $$=\frac{\mathrm{1}}{\mathrm{2}}×\left(\mathrm{1}+{e}\right)×\mathrm{1}=\frac{\mathrm{1}+{e}}{\mathrm{2}}=\mathrm{1}.\mathrm{86} \\$$ $${from}\:{graph}\:... \\$$ $$\mathrm{1}<\int_{\mathrm{0}} ^{\mathrm{1}} {e}^{{x}^{\mathrm{2}} } {dx}<\mathrm{1}.\mathrm{86} \\$$

Commented bytanmay.chaudhury50@gmail.com last updated on 20/Jan/19