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It's been a few years since I became done with my university studies so I don't recall all the details very well. Today, I got into a discussion with my boss, also engineer, regarding computation of an area of a circle (not really so, but that's the significant point).

We tried to set up a double integral as follows:

$A = \iint 1 dr df, r \in [0, R], f \in [0, 2 pi]$

but apparently all the math since a few centuries has been very wrong because we "discovered" that the area is:

$A = \iint dr df = R \int df = 2 \pi R, $

which, of course, is the circumference. :)

My questions are those.

  1. Should I not integrate the function 1 over the specified area?
  2. What did we assume wrongly?
  3. Can I have a hint on how to integrate myself to the correct expression?

(Also, since I realize that I may be jumped with a gazillion of accusation yelling "homework", I can assure that I'm not a student - I can be Googled, Linked-in etc.)

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up vote 7 down vote accepted
  1. You should integrate 1, because $S=\int\limits_S1d\sigma$ In this notation, $d\sigma$ is a "surface element" when integrating over areas (think of calculating a weight of a plate of area S with constant specific density of $1kg/cm^2$ - every little piece of a plate weights $1d\sigma$ kg ).
  2. You assumed that in polar coordinate system $d\sigma=drd\phi$ which is wrong.
  3. Figure out a correct form of $d\sigma$ in polar coordinates, given that in cartesian coordinates it is $d\sigma=dxdy$ and $x=r\cos\phi$, $y=r\sin\phi$
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Splendid reply (answer to the actual questions with exactly corresponding information) BUT too explicit info. I wanted to kill the beast myself... :( Thanks guys, anyway! I've been missing mathematics the last few years... Maybe I should get back and do Ph.D... – Konrad Viltersten Oct 16 '12 at 15:01

You have to take the Jacobian determinant in to account. (Example 3)

Your integral should by $\int_0^{2\pi}\int_0^Rr\,dr\,df$.

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The problem you have is that you defined your integral over two variables, r and f which are not the classical x,y euclidean coordinates. This is a change of variables. You therefore have to add a term in the integral corresponding to the determinant of the Jacobian of this change of variables.

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