Answer
Please see the figure attached.
$\mathop \smallint \limits_{x = 0}^4 \mathop \smallint \limits_{y = 0}^x \cos y{\rm{d}}y{\rm{d}}x = 1 - \cos 4$
Work Step by Step
We have $f\left( {x,y} \right) = \cos y$ and the domain ${\cal D} = \left\{ {0 \le x \le 4,0 \le y \le x} \right\}$.
Considering ${\cal D}$ as a vertically simple region, we evaluate:
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} \cos y{\rm{d}}A = \mathop \smallint \limits_{x = 0}^4 \mathop \smallint \limits_{y = 0}^x \cos y{\rm{d}}y{\rm{d}}x$
$ = \mathop \smallint \limits_{x = 0}^4 \left( {\sin y|_0^x} \right){\rm{d}}x = \mathop \smallint \limits_{x = 0}^4 \sin x{\rm{d}}x$
$ = - \left( {\cos x|_0^4} \right) = 1 - \cos 4$
So, $\mathop \smallint \limits_{x = 0}^4 \mathop \smallint \limits_{y = 0}^x \cos y{\rm{d}}y{\rm{d}}x = 1 - \cos 4$.
