ATAR Notes: Forum

Uni Stuff => Science => Faculties => Mathematics => Topic started by: cara.mel on October 11, 2008, 01:55:23 pm

Title: What is a separable integral
Post by: cara.mel on October 11, 2008, 01:55:23 pm: A lot of people aren't online so I can't ask on irc :(

In my notes for a subject in an example we are doing:
$\int_0^{\frac{\pi}{4}} \int_1^2 \frac{1}{r^3} r dr d\theta$

It then on the next line says this is equal to $=\int_0^{\frac{\pi}{4}} d\theta \times \int_1^2 \frac{1}{r^2} dr$ and a little arrow pointing to a sidenote saying it is a separable integral

I have tried to google it to find out what this is but that didn't tell me anything. So what does it mean, when can you use it.

THANK YOU COBLIN FOR THE LATEX :)
Title: Re: What is a separable integral
Post by: Collin Li on October 11, 2008, 02:01:59 pm: I'm not sure. Here's the LaTeX:

$\int_0^{\frac{\pi}{4}} \int_1^2 \frac{1}{r^3} r dr d\theta$

$=\int_0^{\frac{\pi}{4}} d\theta \times \int_1^2 \frac{1}{r^2} dr$

I guess it's because if you can integrate the inside, it wont affect the outer integral, because there will still be no $\theta$ present, hence you can basically separate it, because the $F(r) = \int_1^2 \frac{1}{r^3} r dr$ integral on the inside is just a constant to the integral with respect to $\theta$ .
Title: Re: What is a separable integral
Post by: cara.mel on October 11, 2008, 02:02:49 pm: thank you :)
Title: Re: What is a separable integral
Post by: humph on October 12, 2008, 01:38:04 am: A multivariable function $F(x,y)$ is separable if it can be written in the form $F(x,y) = X(x)Y(y)$ . In your case above, you had that $F(r,\theta) = R(r)\Theta(\theta)$ with $R(r) = 1/r^2$ and $\Theta(\theta) = 1$ . The basic idea is that
$\iint{F(x,y) \: dxdy} = \iint{X(x)Y(y) \: dxdy} = \int{X(x) \: dx}\int{Y(y) \: dy}$ ,
i.e. you can split the integral up and integrate over each variable separately because they're separable.
Title: Re: What is a separable integral
Post by: cara.mel on October 12, 2008, 08:47:33 am: Thank you very much :)