Multiple integrals help

[lou_skywalker]

the integral (between pi/6 and 0) of the integral (between pi/3 and 0) xsin(x+y)dydx

 

[BigGlenntheHeavy]

\(\displaystyle \int_ {0}^{\pi/6}\int_ {0}^{\pi/3}xsin(x+y)dydx \ = \ \int_{0}^{\pi/6}-xcos(x+y)\bigg]_{0}^{\pi/3}dx\)

\(\displaystyle = \ \int_{0}^{\pi/6}[xcos(x)-xcos(x+\pi/3)]dx \ = \ \frac{1}{2}\int_{0}^{\pi/6}xcos(x)dx+\frac{\sqrt3}{2}\int_{0}^{\pi/6}xsin(x)dx\)

\(\displaystyle = \ .063912395792+.040313620193 \ = \ .104226015985\)

\(\displaystyle Note: \ In \ the \ second \ row, \ expand \ cos(x+\pi/3) \ and \ then \ use \ integration \ by \ parts.\)

\(\displaystyle Afterthought: \ Or, \ if \ you \ have \ a \ trusty \ TI-89, \ just \ plug \ in \ the \ double \ integral \ and, \ walla,\)

\(\displaystyle you \ have \ your \ answer.\)

 

[lou_skywalker]

ok got it now many thanks!