## anonymous one year ago Can someone walk me through how to switch the order of integration on some double integrals?

1. anonymous

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

2. anonymous

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?

3. ganeshie8

Looks good!

4. anonymous

Im kind of confused on what to do next though.

5. ganeshie8

next integrate

6. ganeshie8

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

7. anonymous

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?

8. ganeshie8

Perfect! you may use wolfram to double check http://www.wolframalpha.com/input/?i=%5Cint_0%5E1%5Cint_%7By%2F2%7D%5E%7B1%2F2%7De%5E%7B-x%5E2%7Ddxdy

9. anonymous

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.

10. ganeshie8

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

11. anonymous

Thanks. :)

12. ganeshie8

np:)