Polar coordinate system
Generalization

Using Cartesian coordinates, an infinitesimal area element can be calculated as dA = dx dy. The substitution rule for multiple integrals states that, when using other coordinates, the Jacobian determinant of the coordinate conversion formula has to be considered: J = det ∂ (x, y) ∂ (r, φ) = | ∂ x ∂ r ∂ x ∂ φ ∂ y ∂ r ∂ y ∂ φ | = | cos φ − r sin φ sin φ r cos φ | = r cos 2 φ + r sin 2 φ = r. {\displaystyle J=\det {\frac {\partial (x,y)}{\partial (r,\varphi)}}={\begin{vmatrix}{\frac {\partial x}{\partial r}}&{\frac {\partial x}{\partial \varphi }}\\{\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.}

Hence, an area element in polar coordinates can be written as d A = d x d y = J d r d φ = r d r d φ. {\displaystyle dA=dx\,dy\ =J\,dr\,d\varphi =r\,dr\,d\varphi.}

Now, a function, that is given in polar coordinates, can be integrated as follows: ∬ R f (x, y) d A = ∫ a b ∫ 0 r (φ) f (r, φ) r d r d φ. {\displaystyle \iint _{R}f(x,y)\,dA=\int _{a}^{b}\int _{0}^{r(\varphi)}f(r,\varphi)\,r\,dr\,d\varphi.}

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.

A more surprising application of this result yields the Gaussian integral: ∫ − ∞ ∞ e − x 2 d x = π. {\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}.}
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