Change order of integration and hence evaluate

[DryRun]

Homework Statement
http://s1.ipicture.ru/uploads/20111229/vqL4SkL9.jpg

The attempt at a solution
So, i drew the graph of y against x although i don't know initially if a>b or a<b.
http://s1.ipicture.ru/uploads/20111229/S3y3oSFS.jpg
Looking at the limits, i realized that a must be less than b. So, the shaded areas are the regions which i need to find.

Description:
For y fixed, y varies from y=0 to y=a
and x varies from x=0 to x = (a/b)√(b^2 - y^2)

To change the order of integration, d.w.r.t.y first and then d.w.r.t.x. The region is split into 2 parts.

The region A is the red section on the graph and region B is the blue section on the graph.

Region A:
For x fixed, x varies from x=0 to x=(a/b)√(b^2 - a^2)
and y varies from y=0 to y=a

Region B:
For x fixed, x varies from x = (a/b)√(b^2 - a^2) to x=a
and y varies from y=0 to y=√[b^2 - (b^2.x^2)/a^2]

At this point, i checked but not sure if my work above is correct. But I'm moving on.

Thus, after changing the order, the double integral is equal to the sum of the double integral for region A and B.

I did the calculations in my copybook and the answer has become so complicated. The answer in my notes points to something quite simple.

 

[cragar]

It looks like you are integrating over an ellipse, you could do a change of variables and integrate over a circle. And then factor in the Jacobian.

 

[DryRun]

That's what i did above. There's no need for Jacobian... My notes don't show anything related to Jacobian at this stage. I don't think you read my partial answer.

 

[cragar]

on region A x goes from 0 to what ever that straight vertical lines is, not the equation for that curve. On region B x goes from that vertical line to x=a if your going to do y first.
And when I meant change of variables, right now we have an ellipse
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
we could substitute in R=x/a and Q=y/b and then we would get an equation for a circle and then integrate this in the R-Q plane. Not sure if it will help in this case though

 

[DryRun]

The limits you posted above match the ones i used. The vertical line is obtained by substituting y=a into the equation of the ellipse, as i need the x-coordinate for the point of intersection so as to change the order.
The resulting equation is a bit complex, so i didn't write it on the graph. Here it is:

Replace y = a into $$x = \frac{a}{b}\sqrt (b^2-y^2)$$
Then, $$x = \frac{a}{b}\sqrt (b^2-a^2)$$
So, i have to find $$\int_0^a \! \int_0^\frac{a\sqrt (b^2-y^2)}{b}xy\,dy\,dx$$

For region A:
x fixed
The limits are:
$$0≤x≤\frac{a}{b}\sqrt (b^2-a^2)$$
$$0≤y≤a$$

For region B:
x fixed
The limits are:
$$\frac{a}{b}\sqrt (b^2-a^2)≤x≤a$$
$$0≤y≤\sqrt (b^2-\frac{b^2x^2}{a^2})$$

When evaluating the double integral for region A:
$$\frac{a^4 (b^2-a^2)}{4b^2}$$

My problem is with evaluating the double integral for region B.
I've tried many times and verified my calculations but i can't get the final answer:
http://s1.ipicture.ru/uploads/20111229/rb3LRWcs.jpg
And this tells me that i somehow inexplicably have the graph wrong, and i think in that case a>b or a=b? Can someone please help?

Edit: OK, i see it now. a=b. When y=a, x=0. It is very confusing though and i don't think i would have been able to evaluate this problem without deducing the value of a relative to b, without analysing the answer. Trick question!

 

[HallsofIvy]

Homework Statement
http://s1.ipicture.ru/uploads/20111229/vqL4SkL9.jpg

The attempt at a solution
So, i drew the graph of y against x although i don't know initially if a>b or a<b.
http://s1.ipicture.ru/uploads/20111229/S3y3oSFS.jpg
Looking at the limits, i realized that a must be less than b. So, the shaded areas are the regions which i need to find.

No, they are not. In your limits of integration, x varies from 0 to a. For each x, y varies from 0 to the y value on the ellipse. The integration is over the entire ellipse in the first quadrant. y varies from 0 to b, and for each y, x varies from 0 to $(b/a)a\sqrt{a^2- x^2}$

[quoter]Description:
For y fixed, y varies from y=0 to y=a
and x varies from x=0 to x = (a/b)√(b^2 - y^2)

To change the order of integration, d.w.r.t.y first and then d.w.r.t.x. The region is split into 2 parts.

The region A is the red section on the graph and region B is the blue section on the graph.

Region A:
For x fixed, x varies from x=0 to x=(a/b)√(b^2 - a^2)
and y varies from y=0 to y=a

Region B:
For x fixed, x varies from x = (a/b)√(b^2 - a^2) to x=a
and y varies from y=0 to y=√[b^2 - (b^2.x^2)/a^2]

At this point, i checked but not sure if my work above is correct. But I'm moving on.

Thus, after changing the order, the double integral is equal to the sum of the double integral for region A and B.

I did the calculations in my copybook and the answer has become so complicated. The answer in my notes points to something quite simple.[/QUOTE]