I am trying to figure out how a formula I am looking at was derived. Given a 2-D function $f(x,y)$ that has a constant value of $\rho$ within an ellipse given by
$$ \frac {x^2} {A^2} + \frac {y^2} {B^2} = 1,$$
and is zero outside of this ellipse, the projection of $f(x,y)$ along the line defined by $\delta ( x \cos \theta + y \sin \theta - t )$ is such and such.
Can someone help me set this up? I can find plenty of material on the web about integrating along line segments; however, when I go to calculate the endpoints of of the line segment, the results are way too messy and complicated to be the "correct" way.
Apparently, the answer is
$$\frac {2 \rho A B} {a^2 (\theta) } \sqrt { a^2 (\theta) - t^2 }$$
where
$$a^2 (\theta) = A^2 \cos^2 \theta + B^2 \sin^2 \theta$$
for $|t| \leq a(\theta)$.