# Thread: Reverse order of integration

1. ## Reverse order of integration

Compute the following by reversing the order of integration:

$\displaystyle \int_0^1\int_1^x xye^{y^4} dy dx$

Reversing the order, that's:
$\displaystyle \int_0^1\int_y^1 xye^{y^4} dx dy$

I got to a stage where I have:

$\displaystyle \frac{1}{2}\int_0^1 e^{y^4}(y-y^3) dy$.
Is that correct so far?

I don't know if I can integrate this any further though... Any idea?

2. Hello,

No, there is a slight problem.

Because $\displaystyle x\in[0,1]\Rightarrow x\leq 1$
And y is between 1 and x, which means that $\displaystyle x\leq y\leq 1$

So $\displaystyle 0\leq x\leq y\leq 1$

Hence we have :

$\displaystyle \int_0^1\int_1^x f(x,y) ~dydx=-\int_0^1\int_x^1 f(x,y) ~dydx=-\int_0^1\int_0^y f(x,y) ~dxdy$

that should be easier (and more correct) to deal with !

P.S. : by the way, thank you very much, because in the three threads I've seen from you, you've always showed your work, and that's a very good thing

3. Oops, heh. I was looking at the wrong region. Thanks!

(PS. No problem re: the working. Thanks for noticing)