My answer isGiven define by
Given , calculate the area of the sector of the ellipse defined by .
Is this correct? I'm slightly worried about my r limits...
There's certainly something wrong. Surely the answer should be (a and b must come into it somewhere). But at this time of the evening I don't immediately see how to set up the integral. Maybe my brain will be fresher tomorrow morning.
I just thought of another way of doing it:
and and then substitute these into the equation for the ellipse. This will give an expression for r in terms of and .
In my original post I let , but the integrand would now be the expression I calculated from above.
It's a little late to try this now, i'll try it tomorrow.
Does anyone think this method will work?
The difficulty with this problem is that the notation is misleading. The variables r and θ are not polar coordinates. They refer not to the point , but to the point .
When you transform an integral to polar coordinates, you have to replace dxdy by rdrdθ. The reason for the additional r is that it is the absolute value of the Jacobian determinant of the change of coordinates. For the variables r, θ in this problem, the Jacobian determinant is .
So the integral for the area of that sector of the ellipse is .