How do I change the order of integration of this integral? $$\int\limits_1^e\int\limits_{\frac{\pi}{2}}^{\log \,x} - \sin\,y\, dy\,dx$$
I don't understand how to change the order because of the $\log\,x$ as the upper bound for the inner integral
How do I change it so it looks like $$\int \int f(x)d(y)\,dx\,dy$$
Thanks

Your region is $S=\{(x,y): 1\leq x\leq e,  \log x\leq y\leq \pi/2 \}$. You split it in two regions, $S_1=\{(x,y):1\leq x\leq e, 1\leq y\leq \pi/2\}$ (a rectangle) and $S_2=\{(x,y):0\leq y\leq 1, 1\leq x\leq e^y\}$. Then you write your integral as a sum over each of these regions, and here you have the integration in the order you want. 
