Calculus

Double Integrals

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Double Integrals in Polar Coordinates

One of the particular cases of change of variables is the transformation from Cartesian to polar coordinate system \(\left({\text{Figure }1}\right):\)

\[x = r\cos \theta ,\;\;y = r\sin \theta .\]
Polar coordinate system
Figure 1.

The Jacobian determinant for this transformation is

\[ \frac{{\partial \left( {x,y} \right)}}{{\partial \left( {r,\theta } \right)}} = \left| {\begin{array}{*{20}{c}} {\frac{{\partial x}}{{\partial r}}}&{\frac{{\partial x}}{{\partial \theta }}}\\ {\frac{{\partial y}}{{\partial r}}}&{\frac{{\partial y}}{{\partial \theta }}} \end{array}} \right| = \left| {\begin{array}{*{20}{c}} {\frac{{\partial \left( {r\cos \theta } \right)}}{{\partial r}}}&{\frac{{\partial \left( {r\cos \theta } \right)}}{{\partial \theta }}}\\ {\frac{{\partial \left( {r\sin \theta } \right)}}{{\partial r}}}&{\frac{{\partial \left( {r\sin \theta } \right)}}{{\partial \theta }}} \end{array}} \right| = \left| {\begin{array}{*{20}{c}} {\cos \theta }&{ - r\sin \theta }\\ {\sin\theta }&{r\cos \theta } \end{array}} \right| = \cos \theta \cdot r\cos \theta - \left( { - r\sin \theta } \right) \cdot \sin \theta = r\,{\cos ^2}\theta + r\,{\sin ^2}\theta = r\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right) = r.\]

As a result, the differential for polar coordinates is

\[dxdy = \left| {\frac{{\partial \left( {x,y} \right)}}{{\partial \left( {r,\theta } \right)}}} \right|drd\theta = rdrd\theta .\]

Let the region \(R\) in polar coordinates be defined as follows (Figure \(2\)):

\[0 \le g\left( \theta \right) \le r \le h\left( \theta \right),\;\; \alpha \le \theta \le \beta ,\;\; \text{where}\;\;\beta - \alpha \le 2\pi .\]
A region of integration in polar coordinates
Figure 2.
Polar rectangle
Figure 3.

Then the double integral in polar coordinates is given by the formula

\[\iint\limits_R {f\left( {x,y} \right)dxdy} = \int\limits_\alpha ^\beta {\int\limits_{g\left( \theta \right)}^{h\left( \theta \right)} {f\left( {r\cos \theta ,r\sin \theta } \right)rdrd\theta } } \]

The region of integration (Figure \(3\)) is called the polar rectangle if it satisfies the following conditions:

\[0 \le a \le r \le b,\;\; \alpha \le \theta \le \beta ,\;\; \text{where}\;\;\beta - \alpha \le 2\pi .\]

In this case the formula for change of variables can be written as

\[\iint\limits_R {f\left( {x,y} \right)dxdy} = \int\limits_\alpha ^\beta {\int\limits_{a}^{b} {f\left( {r\cos \theta ,r\sin \theta } \right)rdrd\theta } } .\]

Be careful not to forget the factor \(r\) (the Jacobian) in the right-hand side of the formula!

Solved Problems

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Example 1

Calculate the double integral \[\iint\limits_R {\left( {{x^2} + {y^2}} \right)dydx}\] by transforming to polar coordinates. The region of integration \(R\) is the sector \(0 \le \theta \le {\frac{\pi }{2}}\) of a circle with radius \(r = \sqrt 3.\)

Example 2

Evaluate the integral \[\iint\limits_R {xydydx},\] where the region of integration \(R\) lies in the sector \(0 \le \theta \le \frac{\pi}{2}\) between the curves \({x^2} + {y^2} = 1\) and \({x^2} + {y^2} = 5.\)

Example 1.

Calculate the double integral \[\iint\limits_R {\left( {{x^2} + {y^2}} \right)dydx}\] by transforming to polar coordinates. The region of integration \(R\) is the sector \(0 \le \theta \le {\frac{\pi }{2}}\) of a circle with radius \(r = \sqrt 3.\)

Solution.

The region \(R\) is the polar rectangle \(\left({\text{Figure }4}\right)\) and described by the set

\[R = \Big\{ {\left( {r,\theta } \right)|\;0 \le r \le \sqrt 3 ,\; 0 \le \theta \le {\frac{\pi }{2}}} \Big\} .\]
Region of integration as a sector of a circle
Figure 4.

Using the formula

\[\iint\limits_R {f\left( {x,y} \right)dxdy} = \int\limits_\alpha ^\beta {\int\limits_{a}^{b} {f\left( {r\cos \theta ,r\sin \theta } \right)rdrd\theta } },\]

we obtain

\[\iint\limits_R {\left( {{x^2} + {y^2}} \right)dydx} = \int\limits_0^{\frac{\pi }{2}} {\int\limits_0^{\sqrt 3 } {{r^2}\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right)rdrd\theta } } = \int\limits_0^{\frac{\pi }{2}} {d\theta } \int\limits_0^{\sqrt 3 } {{r^3}dr} = \left. \theta \right|_0^{\frac{\pi }{2}} \cdot \left. {\left( {\frac{{{r^4}}}{4}} \right)} \right|_0^{\sqrt 3 } = \frac{\pi }{2} \cdot \frac{9}{4} = \frac{{9\pi }}{8}.\]

Example 2.

Evaluate the integral \[\iint\limits_R {xydydx},\] where the region of integration \(R\) lies in the sector \(0 \le \theta \le \frac{\pi}{2}\) between the curves \({x^2} + {y^2} = 1\) and \({x^2} + {y^2} = 5.\)

Solution.

In polar coordinates, the region of integration \(R\) is the polar rectangle \(\left({\text{Figure }5}\right):\)

\[R = \left\{ {\left( {r,\theta } \right)|\;1 \le r \le \sqrt 5 ,\; 0 \le \theta \le \frac{\pi}{2} } \right\}.\]
The region of integration is a ring in the first quadrant
Figure 5.

So using the formula

\[\iint\limits_R {f\left( {x,y} \right)dxdy} = \int\limits_\alpha ^\beta {\int\limits_{a}^{b} {f\left( {r\cos \theta ,r\sin \theta } \right)rdrd\theta } },\]

we find the integral:

\[\iint\limits_R {xydydx} = \int\limits_0^{\frac{\pi}{2}} {\int\limits_1^{\sqrt 5 } {r\cos \theta r\sin \theta rdrd\theta } } = \int\limits_0^{\frac{\pi}{2}} {\sin \theta \cos \theta d\theta } \int\limits_1^{\sqrt 5 } {{r^3}dr} = \frac{1}{2}\int\limits_0^{\frac{\pi}{2}} {\sin 2\theta d\theta } \int\limits_1^{\sqrt 5 } {{r^3}dr} = \frac{1}{2}\left. {\left( { - \frac{{\cos 2\theta }}{2}} \right)} \right|_0^{\frac{\pi}{2}} \cdot \left. {\left( {\frac{{{r^4}}}{4}} \right)} \right|_1^{\sqrt 5 } = \frac{1}{4}\left( { - \cos \pi + \cos 0} \right) \cdot \frac{1}{4}\left( {25 - 1} \right) = \frac{1}{4}\left( { 1 + 1} \right) \cdot 6 = 3.\]

See more problems on Page 2.

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