# If R=[0,1]x [0,1], show that 0le int int_R sin(x+y)dA le 1

If R=[0,1]x [0,1], show that $0\le \int {\int }_{R}\mathrm{sin}\left(x+y\right)dA\le 1$
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

phravincegrln2
If R = [0, 1] x [0, 1], show that $0\le \int {\int }_{R}\mathrm{sin}\left(x+y\right)dA\le 1$
the region $R=\left(\left(x,y\right)|0\le x\le 1,0\le y\le 1\right)$
therefore the integral becomes.
${\int }_{0}^{1}{\int }_{0}^{1}\mathrm{sin}\left(x+y\right)dxdy$
$={\int }_{0}^{1}\left[-\mathrm{cos}\left(x+y\right){\right]}_{0}^{1}dy$
$=-{\int }_{0}^{1}\left[\mathrm{cos}\left(1+y\right)-\mathrm{cos}\left(y\right)\right]dy$
$=-\left[\mathrm{sin}\left(1+y\right)-\mathrm{sin}\left(y\right){\right]}_{0}^{1}$
$=-\left[\left(\mathrm{sin}2-\mathrm{sin}1\right)-\left(\mathrm{sin}1-\mathrm{sin}0\right)\right]$
$=2\mathrm{sin}1-\mathrm{sin}2\cong 0.774$
$0\le 0.774\le 1$