Double Integrals: How Do I Integrate a Quotient Function Over a Region?

[Heat]

Homework Statement

Finding the double integral of the following

$$\int\int xy / (x^2+y^2+1)^1/2 dA$$

R = [(x,y): 0<=x<=1, 0<=y<=1]

Homework Equations

None

The Attempt at a Solution

ok I am having trouble integrating when I see the the quotient.

what I have done is,

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

I can't remember the step taken to ingrate this.

I would add one to the exponent making it,

(2)xy(x^2+y^2+1)^1/2 <---but I am missing something else. I would need to integrate each y right?

so that would be

$$\int x(x^2+ (1/3)y^3 +1 ) ^1/2 dx$$

and then I proceed integrating it again, this time for x... ?

 

[Mark44]

The stuff you put in LaTex isn't formatting, so I'll have to go with what I think the problem is.

You can change from the double integral over the region R to iterated integrals either by integrating with respect to y first and then x, or by integrating with respect to x first and then y. The region R is pretty simple, which makes matters simpler.

Using the same order that you chose, the inner integral is from y = 0 to y = 1, and the integrand is xy/(x^2 + y^2 + 1)^(1/2), and you're integrating with respect to y.

You can pull the factor of x outside the inner integral, since you're integrating w.r.t. y and x is constant on this strip.

A straightforward substitution will work: u = x^2 + y^2 + 1, so du = 2ydy. Remember, x is not varying, so x can be considered to be a constant.

Your integrand now looks like 1/2 u^(-1/2) du, and its antiderivative is 1/2 * 2 * u^(1/2) = u^(1/2).

Undoing the substitution gives (x^2 + y^2 + 1)^(1/2).

Evaluate the expression above at y = 1 and at y = 0, and subtract the second value from the first. This will leave you with a function of x alone.

Now, integrate the resulting function w.r.t. x and evaluate its antiderivative at x = 1 and at x = 0, and subtract the second expression from the first.

Don't forget that you pulled a factor of x out of the first integral; you'll need to incorporate this into the outer integrand.

Hope that's enough to get you going.