Question Number 26399 by abdo imad last updated on 25/Dec/17
$${calculate}\:\:\int\int\:_{{D}} {cos}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right){dxdy}\:\:\:{with}\:\:{D}={C}\left({o}.\sqrt{\frac{\pi}{\mathrm{2}}}\right). \\ $$
Answered by kaivan.ahmadi last updated on 25/Dec/17
Answered by ajfour last updated on 25/Dec/17
$$=\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \int_{\mathrm{0}} ^{\:\:\sqrt{\frac{\pi}{\mathrm{2}}}} \left(\mathrm{cos}\:{r}^{\mathrm{2}} \right)\left({rdr}\right){d}\theta \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{if}\:\:{r}^{\mathrm{2}} ={x}^{\mathrm{2}} +{y}^{\mathrm{2}} \\ $$$$=\pi\int_{\mathrm{0}} ^{\:\:\sqrt{\pi/\mathrm{2}}} \left(\mathrm{cos}\:{r}^{\mathrm{2}} \right)\left(\mathrm{2}{rdr}\right) \\ $$$$=\pi\left(\mathrm{sin}\:{r}^{\mathrm{2}} \right)\mid_{\mathrm{0}} ^{\sqrt{\pi/\mathrm{2}}} \:=\:\boldsymbol{\pi}\:. \\ $$