Can someone walk me through how to switch the order of integration on some double integrals?

Question

anonymous · Answer

$$\text{Original integral:}~\int_0^1\int_{y/2}^{1/2}e^{-x^2}dxdy$$

anonymous · Answer

I tried changing the bounds, and I got $$\int_0^{1/2}\int_0^{2x}f(x,y)dydx$$Am I on the right track so far?

ganeshie8 · Answer

Looks good!

anonymous · Answer

Im kind of confused on what to do next though.

ganeshie8 · Answer

next integrate

ganeshie8 · Answer

start with the inner integral :

 $$\large \int_0^{1/2} \color{blue}{\int_0^{2x}e^{-x^2}dy}~dx$$

anonymous · Answer

OH! I forgot that I can actually integrate it now. So, it becomes:
$$\int_0^{1/2}2xe^{-x^2}dx=-(e^{-x^2})|_0^{1/2}=1-e^{-1/4}$$Is that right?

ganeshie8 · Answer

anonymous · Answer

Alright, thanks! I thought I had to do something to the integrand, and I didn't recognize that with the change to dydx, I could actually integrate.

ganeshie8 · Answer

Thats it! changing order of integration gave us that factor 2x  which was useful in u-substitution

anonymous · Answer

Thanks. :)

ganeshie8 · Answer

np:)