How Do You Calculate the Area of an Ellipse from Pericenter to Pi/2?

[cfin3]

Homework Statement

I have an orbital mechanics problem, in which I need to find the area of the ellipse from pericenter to pi/2 (semi-latus rectum location).

Homework Equations

So I have the orbit equation
r = p/(1+e*cos(theta)) where the origin is at the focus. So I know that A = integral( 1/2 * r^2 * d_theta) from 0-> pi/2.

The Attempt at a Solution

The problem is when plugging in r I can't integrate it. I need to account for the fact that 0=<e=<1 also. I have read many many other posts and searched on this topic but haven't found a concise answer.


Thanks for the help!

 

[gneill]

I know it may sound unintuitive, by you might want to try performing the integration in a Cartesian form.

Given a semi-latus rectum p and eccentricity e, the semi major and minor axes are:

a = p/(1 - e²) and b = sqrt(ap)

The distance from the center of the ellipse to the focus is sqrt(a² - b²).

The equation of the ellipse in Cartesian form is x²/a² + y²/b² = 1 .

Then follow the example shown http://math.ucsd.edu/~wgarner/math10b/area_ellipse.htm" with the appropriate domain for x.