Evaluate $ \int_{0}^{1} \int_{0}^{\arccos x}e^{\sin x} \,dx\,dy $.
By inspection, $ \int e^{\sin x}\,dx $ is not possible or is difficult. My initial plan was to change the order of integration so that $e^{\sin x}$ will just be treated as a constant, I represented the integral as $$ \int \int_{A} e^{\sin x} \,dy\,dx .$$ However, I am having trouble with how to get the new basic region I denoted as $A$.