# Thread: Changing the order of integration

1. ## Changing the order of integration

Hey I'm having some trouble with multiple integrals if anyone would care to explain (I think I'm just not using the correct region!)

Evaluate
$\int_{x=0}^2\int_{y=\frac{x}{2}}^1 2xy^2 dy dx$

first directly then changing the order of integration
I did the first part and got 0.8 as the answer

just for the second I'm having trouble

I think the surface being integrated over the the triangle bounded by the lines y=x/2 x=0 and y=1 but I may be wrong which gives

$\int_{y=0}^1\int_{x=2y}^2 2xy^2 dx dy$

but the value of that doesnt agree, can anyone explain how to do the limits correctly?

cheers

Simon

2. $

\int\limits_0^1 {\int\limits_0^{x = 2y} {2xy^2 dxdy} }

$

3. Originally Posted by thelostchild
Hey I'm having some trouble with multiple integrals if anyone would care to explain (I think I'm just not using the correct region!)

I did the first part and got 0.8 as the answer

just for the second I'm having trouble

I think the surface being integrated over the the triangle bounded by the lines y=x/2 x=0 and y=1 but I may be wrong which gives

$\int_{y=0}^1\int_{x=2y}^2 2xy^2 dx dy$

but the value of that doesnt agree, can anyone explain how to do the limits correctly?

cheers

Simon
$0\leqslant{x}\leqslant{2}$ and $\frac{x}{2}\leqslant{y}\leqslant{1}\implies{x\leqs lant{2y}\leqslant{2}}$. Stringing these two inequalites together gives

$0\leqslant{x}\leqslant{2y}\leqslant{2}$

From there Peritus's answer should be more apparent.

4. Originally Posted by Mathstud28

$0\leqslant{x}\leqslant{2}$ and $\frac{x}{2}\leqslant{y}\leqslant{1}\implies{x\leqs lant{2y}\leqslant{2}}$. Stringing these two inequalites together gives

$0\leqslant{x}\leqslant{2y}\leqslant{2}$

From there Peritus's answer should be more apparent.
But be careful 'cause, this method not always works.

5. Originally Posted by Krizalid
But be careful 'cause, this method not always works.
Really? I have never encountered a case where it hasn't. Would you mind giving me an example of one please?

6. Try it with $\int_0^1\int_{x^3}^{\sqrt[3]x}dy\,dx.$

7. Originally Posted by Krizalid
Try it with $\int_0^1\int_{x^3}^{\sqrt[3]x}dy\,dx.$
Thank you, I will report back later when I have had time to look at it.

8. Also be careful that reversing integration order is submitted to the condition that $\int \int \left|f(x,y)\right| ~ dx ~ dy$ is a finite value.
See Fubini's theorem - Wikipedia, the free encyclopedia for further information (advanced calculus imo)