Editing Polar coordinate system (section)

====Generalization====
Using [[Cartesian coordinates]], an infinitesimal area element can be calculated as ''dA'' = ''dx'' ''dy''. The [[integration by substitution#Substitution for multiple variables|substitution rule for multiple integrals]] states that, when using other coordinates, the [[Jacobian matrix and determinant|Jacobian]] determinant of the coordinate conversion formula has to be considered:
<math display="block">J = \det \frac{\partial(x, y)}{\partial(r, \varphi)}
  = \begin{vmatrix}
      \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \varphi} \\[2pt]
      \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \varphi}
    \end{vmatrix}
  = \begin{vmatrix}
      \cos\varphi & -r\sin\varphi \\
      \sin\varphi &  r\cos\varphi
    \end{vmatrix}
  = r\cos^2\varphi + r\sin^2\varphi = r.
</math>

Hence, an area element in polar coordinates can be written as
<math display="block">dA = dx\,dy\ = J\,dr\,d\varphi = r\,dr\,d\varphi.</math>

Now, a function, that is given in polar coordinates, can be integrated as follows:
<math display="block">\iint_R f(x, y)\, dA = \int_a^b \int_0^{r(\varphi)} f(r, \varphi)\,r\,dr\,d\varphi.</math>

Here, ''R'' is the same region as above, namely, the region enclosed by a curve ''r''(''φ'') and the rays ''φ'' = ''a'' and ''φ'' = ''b''. The formula for the area of ''R'' is retrieved by taking ''f'' identically equal to 1.

[[Image:E^(-x^2).svg|thumb|right|A graph of <math>f(x) = e^{-x^2}</math> and the area between the function and the <math>x</math>-axis, which is equal to <math>\sqrt{\pi}</math>.]]

A more surprising application of this result yields the [[Gaussian integral]]:
<math display="block">\int_{-\infty}^\infty e^{-x^2} \, dx = \sqrt\pi.</math>