Find the double integral

**Rhode963** · March 2nd 2010, 06:58 AM

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 $(xy((x^2+y^2)^(1/2)))$ dy)dx

**TheEmptySet** · March 2nd 2010, 07:15 AM

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 $(xy((x^2+y^2)^(1/2)))$ dy)dx

I think this is what you mean

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

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

This is exactly like integrating

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

I hope this helps.

**Rhode963** · March 2nd 2010, 07:18 AM

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

**Rhode963** · March 2nd 2010, 07:29 AM

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

**TheEmptySet** · March 2nd 2010, 07:31 AM

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 $sqrt(u)du$ ?

It doesn't change anything the substituion still works.

**Rhode963** · March 2nd 2010, 07:35 AM

So for the inside integral, I got the answer to be $(3/2)x$ , 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.

**General** · March 2nd 2010, 12:09 PM

Originally Posted by Rhode963

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

$\int_0^1 \int_0^1 xy \, \sqrt{x^2+y^2} dy \, dx =\int_0^1 x \int_0^1 y \, \sqrt{x^2+y^2} dy \, dx$ .

Just apply the suggested substitution on the inner integral.

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