# Thread: Find the double integral

1. ## Find the double integral

So I'm confused on how to take the first integral because of the radical... please explain how you came to your answer. (S=integral, so S:0,1 means the integral from 0 to 1, just to explain my notation)

S:0,1(S:0,1$\displaystyle (xy((x^2+y^2)^(1/2)))$dy)dx

2. Originally Posted by Rhode963
So I'm confused on how to take the first integral because of the radical... please explain how you came to your answer. (S=integral, so S:0,1 means the integral from 0 to 1, just to explain my notation)

S:0,1(S:0,1$\displaystyle (xy((x^2+y^2)^(1/2)))$dy)dx
I think this is what you mean

$\displaystyle \int_{0}^{1}\int_{0}^{1}\frac{xy}{\sqrt{x^2+y^2}}d ydx$

It is just a u sub. let $\displaystyle u=x^2+y^2 \implies du=2ydy$ Remember x is constant with respect to the inside integral.

This is exactly like integrating

$\displaystyle \int_{0}^{1}\frac{3y}{\sqrt{3^2+y^2}}dy$
where three is playing the role of x!

I hope this helps.

3. Great help, but it is actually the double integral of $\displaystyle xy*sqrt(x^2+y^2)$ from 0 to 1. Sorry, I don't know how to get the integral symbol to work.

4. Would u-sub still work? So for the inner integral I could pull out an x, and get x times the integral of $\displaystyle sqrt(u)du$?

5. Originally Posted by Rhode963
Would u-sub still work? So for the inner integral I could pull out an x, and get x times the integral of $\displaystyle sqrt(u)du$?
It doesn't change anything the substituion still works.

6. So for the inside integral, I got the answer to be $\displaystyle (3/2)x$, does that sound right to you? It seems like a strange amount of canceling. And when plugging into the final integral, I got the final answer to be 3.

7. Originally Posted by Rhode963
Great help, but it is actually the double integral of $\displaystyle xy*sqrt(x^2+y^2)$ from 0 to 1. Sorry, I don't know how to get the integral symbol to work.
$\displaystyle \int_0^1 \int_0^1 xy \, \sqrt{x^2+y^2} dy \, dx =\int_0^1 x \int_0^1 y \, \sqrt{x^2+y^2} dy \, dx$.

Just apply the suggested substitution on the inner integral.