Double integrals over region (xy^2)/(x^2)+y

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kilroymcb

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I'm having a little trouble with this problem, conceptually.

Double integral of the region (xy^2)/(x^2)+y. The limits are from 0-1 with respect to x and -3 - 3 with respect to y. Assuming I take the integral withrespect to y first... Would the best way to do this be using integration by parts?
 
Is this your integral:

\(\displaystyle \L\\\int_{-3}^{3}\int_{0}^{1}\frac{xy^{2}}{x^{2}+y}dxdy\)
 
galactus said:
Is this your integral:

\(\displaystyle \L\\\int_{-3}^{3}\int_{0}^{1}\frac{xy^{2}}{x^{2}+y}dxdy\)
Click to expand...

The order of the limits of integration is wrong... but otherwise, yes.
 
Pull out the x.

\(\displaystyle \L\;\int{x\int{\frac{y}{x^{2}+y}}}\;dy\;dx\)

Consider the partial fraction decomposition:

\(\displaystyle \L\;\int{x\int{\frac{x^{5}}{x^{2}+y}-x^{3}+xy}}\;dy\;dx\)

Is it getting easier or harder?
 
Here's the first portion. You can tackle the second. It's more precarious.

It would seem this integral results in a complex result.

It's been a while since I evaluated a double integral by hand. Too easy to use a calculator these days.

\(\displaystyle \L\\\int\frac{xy^{2}}{x^{2}+y}dy\)

\(\displaystyle \L\\x\int\frac{y^{2}}{x^{2}+y}dy\)

Rewrite:

\(\displaystyle \L\\x\int[-x^{2}+y+\frac{x^{4}}{x^{2}+y}]dy\)

\(\displaystyle \L\\x[-\int{x^{2}}dy+\int{y}dy+\int\frac{x^{4}}{x^{2}+y}dy]\)

\(\displaystyle \L\\x[-x^{2}y+\frac{y^{2}}{2}+x^{4}\int\frac{1}{x^{2}+y}dy]\)

Let \(\displaystyle u=x^{2}+y, \;\ du=dy\)

\(\displaystyle \L\\x[-x^{2}y+\frac{y^{2}}{2}+x^{4}\int\frac{1}{u}du]\)

\(\displaystyle \L\\x[-x^{2}y+\frac{y^{2}}{2}+x^{4}ln(x^{2}+y)]\)

Using the limits of integration, -3..3, we get:

\(\displaystyle \L\\x^{5}ln(x^{2}+3)-x^{5}ln(x^{2}-3)-6x^{3}\)

You can tackle the wrt x part. Parts would come in handy now.
 
I am confused here.

Why are you integrating w.r.t. 'y' first.

If you integrate w.r.t. 'x' first - pulling out y^2 - the last integral becomes

1/2 * y^2[ln(1+y) - ln(y)] dy

These are lot easier to handle.

I have not done these for a while - am I making some 'fundamental' mistake??
 
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