Integrating the generator lines of an elliptical orbit

[Appleton]

Hi,
I am having difficulty understanding the following:

\int^{2π}_{0}(x+y)\,dθ = \int^{2π}_{0} 2a\,dθ = \textbf{4}πa

where x and y are the generator lines of an elipse, a is the semimajor axis and θ is the angle formed by x and the major axis.

I understand that x+y = 2a. However I don't understand how to integrate with respect to θ without expressing (x+y) or 2a in terms of θ.

Any help would be much appreciated.

 

[HallsofIvy]

The integral of a constant, such as 2a, with respect to any differential, dx, is that constant times the variable x. Normally, \int C dx= Cx (plus a constant of integration) is one of the first integral facts one learns in Calculus.

 

[scurty]

Hi,
I am having difficulty understanding the following:

\int^{2π}_{0}(x+y)\,dθ = \int^{2π}_{0} 2a\,dθ = \textbf{4}πa

where x and y are the generator lines of an elipse, a is the semimajor axis and θ is the angle formed by x and the major axis.

I understand that x+y = 2a. However I don't understand how to integrate with respect to θ without expressing (x+y) or 2a in terms of θ.

Any help would be much appreciated.

Because 2a and (x + y) are not the variables of integration, they are treated as constants and can thus be pulled out of the integral.

$\displaystyle \int_{0}^{2 \pi} 2a \ d\theta = 2a \int_{0}^{2 \pi} \ d\theta = \Bigl.2a \theta \ \Bigr|_{\theta = 0}^{\theta = 2 \pi} = 4 \pi a$

 

[Appleton]

Thanks for the help. I stumbled into this whilst reading "Gamma, exploring Euler's constant". I was inspired to get this after reading John Derbyshire's "Unknown Quantity, a real and imagined history of Algebra" which I found hugely illuminating. However as someone who only studied maths until 16 I feel I may have bitten off more than I can chew with this new book.