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Double integral in Polar coordinates

Besides rectangular coordinates, it is convenient to consider polar coordinates on ℝ². They oridinates in the work by the ancient Greek astronomer and astrologer Hipparchus (190–120 BC). The full history of the subject is described in Harvard professor Julian Lowell Coolidge's Origin of Polar Coordinates.

In order to introduce polar coordinates, start with a point O in the plane ℝ² called the pole (we will always identify this point with the origin). From the pole, draw a ray, called the initial ray (we will always draw this ray horizontally, identifying it with the positive abscissa). A point P in the plane is determined by the distance r that P is from O, and the angle θ formed between the initial ray and the segment \( \displaystyle \quad \overline{OP} \quad \) (measured counter-clockwise). We record the distance and angle as an ordered pair (r, θ). This is illustrated in the following figure:

Polar coordinates.
           
point = Graphics[{PointSize[0.02], Pink, Point[{1/2, Sqrt[3]/2}]}]; or = Graphics[{PointSize[0.02], Blue, Point[{0, 0}]}]; xar = Graphics[{Blue, Arrowheads[0.1], Arrow[{{0, 0}, {1, 0}}]}]; line = Graphics[{Black, Thick, Line[{{0, 0}, {1/2, Sqrt[3]/2}}]}]; txt = Graphics[{Black, Text[Style["O", FontSize -> 18, Bold], {0, -0.5}], Text[Style["r", FontSize -> 18, Bold], {0.2, 0.5}], Text[Style["P = P(r, \[Theta])", FontSize -> 18, Bold], {0.5, 0.94}], Text[ Style["\[Theta]", FontSize -> 18, Bold], {0.56, 0.45}]}]; circle = Graphics[Circle[{0, 0}, 0.6, {0, Pi/3}]]; ar = Graphics[{Black, Arrowheads[0.04], Arrow[{{0.32, 0.5059}, {0.3, 0.519}}]}]; Show[txt, xar, line, point, or, circle, ar]

Conversion between rectangular and polar coordinates are based of the relations:

\[ \begin{split} x &= r\,\cos\theta , \\ y &= r\,\sin\theta , \qquad 0 \le r <\infty , \quad 0 \le \theta < 2\pi . \end{split} \]
and
\[ r = +\sqrt{x^2 + y^2} , \qquad \tan\theta = \frac{y}{x} . \]
Defining a new coordinate system allows us to create a new kind of function, a polar function. Rectangular coordinates lent themselves well to creating functions that related x and y, such as y = sinx. Polar coordinates allow us to create functions that relate r and θ. Normally these functions look like r = f(θ), although we can create functions of the form θ = g(r).

The basic form of the double integral is:

\[ \iint_R F\,{\text d}A , \]
which can be interpreted as a Riemann integral over some region, sum up products of heights and areas. Of course if you want to evaluate the integral (and honestly, who doesn’t?) you have to change R to a region defined in (x, y)-coordinates, and change dA to dxdy or dydx leaving iterated integrals
\[ \int_a^b {\text d}x \int_c^d {\text d}y\, F(x, y) \quad \mbox{and} \quad \int_c^d {\text d}y \int_a^b {\text d}x \, F(x, y) . \]
Now consider representing a region R with polar coordinates.

RJB figure https://xronos.clas.ufl.edu/mooculus/calculus3/commonCoordinates/digInPolarCoordinates

Let R be the region in the first quadrant bounded by the curve. We can approximate this region using the natural shape of polar coordinates: Portions of sectors of circles. In the figure, one such region is shaded, shown below:

RJB figure https://xronos.clas.ufl.edu/mooculus/calculus3/commonCoordinates/digInPolarCoordinates

From the picture above, we see that:

\begin{align*} {\text d}A &= {\text d}r \left( r{\text d}\theta \right) \\ &= r\,{\text d}r \,{\text d}\theta . \end{align*}
Recalling that the determinant of a 2×2 matrix gives the area of a parallelogram, we could also deduce the correct formula for dA by setting
\begin{align*} x (r, \theta ) &= r\,\cos\theta , \\ y (r, \theta ) &= r\,\sin\theta \end{align*}
and computing the \[ {\text d}A = \left\vert \det \begin{bmatrix} \frac{\partial x}{\partial r}\,{\text d}r & \frac{\partial y}{\partial r}\,{\text d}r \\ \frac{\partial x}{\partial \theta}\,{\text d}\theta & \frac{\partial y}{\partial \theta}\,{\text d}\theta \end{bmatrix} \right\vert = r\,{\text d}r \,{\text d}\theta . \]
RJB to calculate jacobian
So to evaluate the double integral, we replace dA with rdrdθ and convert the function z = F(x, y) to a function of polar coordinates:
\[ F(r\,\cos\theta , r\,\sin\theta ) . \]
Finally, find bounds g₁(θ) ≤ r ≤ g₂(θ) and α ≤ θ ≤ β that describe R.

Theorem 1: Let F : ℝ² → ℝ be continuous on the region \[ R = \left\{ (r, \theta ) \,: \,\alpha \le \theta \le \beta , \quad g_1 (\theta ) \le r \le g_2 (\theta ) \right\} . \] Then \[ \iint_R F(x,y) \,{\text d}A = \int_{\alpha}^{\beta} {\text d} \theta \int_{g_1 (\theta )}^{g_2 (\theta )} r{\text d}r\,F\left( r\,\cos\theta , r\,\sin]theta \right) . \]
   
Example 1: Compute the area of the lemniscate given by: \[ r = +\sqrt{\cos (2\theta )} , \qquad \theta \in [0, 2\pi] . \] RJB: https://xronos.clas.ufl.edu/mooculus/calculus3/commonCoordinates/digInPolarCoordinates    ■
End of Example 1
   
Example 2:    ■
End of Example 2

 

 

  1. Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
  2. Beezer, R., A First Course in Linear Algebra, 2015.
  3. Beezer, R., A Second Course in Linear Algebra, 2013.
  4. Coolidge, J.L., The Origin of Polar Coordinates, American Mathematical Monthly. 1952, volume 59, issue 2, pp. 78–85. doi:10.2307/2307104.