Author Topic: What is a separable integral (Read 1378 times) Tweet Share

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

Collin Li · « **Reply #1 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 $\theta$ .

cara.mel · « **Reply #2 on:** October 11, 2008, 02:02:49 pm »

thank you

humph · « **Reply #3 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.

cara.mel · « **Reply #4 on:** October 12, 2008, 08:47:33 am »

Thank you very much

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Author Topic: What is a separable integral (Read 1378 times) Tweet Share

cara.mel

What is a separable integral

Collin Li

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cara.mel

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humph

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