How do I find the limits when trying to integrate over an ellipse? (1) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
Edit: I'm trying to find the area of the part of the plane $Ax + By +Cz = D$ lying inside the elliptical cylinder (1). I tried using parameterization for the plane to get the $ r(u,v)$ = ui + vj + $\frac{(D-Au-Bv)}{C}$ k with the normal n(u,v) = $\frac{A}{C}$i + $\frac{B}{C}$j + k.
The area of the intersection between plane and cylinder is the area of the ellipse on plane $xy$ divided by $\cos\theta$, where $\theta$ is the inclination of the given plane with respect to plane $xy$:
$$ A={\pi ab\over\cos\theta}=\pi ab{\sqrt{A^2+B^2+C^2}\over C}. $$
The figure below shows how the second ellipse is a dilation of the first one, by a factor $1/\cos\theta$, along a direction perpendicular to the intersection between the planes (green line).