3.5 Double Integrals In Polar Coordinates

Suppose that we want to integrate \(\displaystyle {\iint \limits _R f(x,y)\hspace {0.1cm}dA}\) where \(R\) is one of the regions shown below.

(a)

\(\big \{ (r,\theta ) \hspace {0.1cm} \big |\hspace {0.1cm} 0 \leq r\leq 1\hspace {0.1cm}, \hspace {0.1cm} 0 \leq \theta \leq 2\pi \big \}\)

(b)

\(\big \{ (r,\theta ) \hspace {0.1cm} \big |\hspace {0.1cm} 1 \leq r\leq 2\hspace {0.1cm}, \hspace {0.1cm} 0 \leq \theta \leq \pi \big \}\)

In either case, the description of \(R\) in rectangular coordinates is rather complicated but \(R\) is easily described in polar coordinates.

\[R = \big \{ (r,\theta ) \hspace {0.1cm} \big | \hspace {0.1cm}a\leq r\leq b\hspace {0.1cm}, \hspace {0.1cm} \alpha \leq \theta \leq \beta \big \}\] divide \([a,b]\) into \(m\) sub intervals and \([\alpha , \beta ]\) into \(n\) subintervals. The circles \(r=r_i\) and the ranges \(\theta = \theta _j\) divide the polar rectangle \(R\) into the smaller polar rectangles \[R_{ij} = \big \{(r,\theta )\hspace {0.1cm}\big |\hspace {0.1cm} r_{i-1} \leq r \leq r_j\hspace {0.1cm} , \hspace {0.1cm} \theta _{j-1}\leq \theta \theta _j\big \}\]

has polar coordinates \(\big (r^*_i, \theta ^*_j\big )\).

The area of \(R_{ij}\) is \begin {align*} \Delta A_i & = \frac {1}{2}r^2_i\Delta \theta - \frac {1}{2}r^2_{i-1}\Delta \theta \\ & = \underbrace {\frac {1}{2}\big (r_i + r_{i-1}\big )}_{r^*_i} \big (r_i - r_{i-1}\big )\Delta \theta \\ & = r^*_{i} \Delta r \Delta \theta \\\\ \end {align*}

Riemann sum \[\sum ^m_{i=1}\sum ^n_{j=1} f\big (r^*_i\cos \theta , r^*_i\sin \theta \big ) \Delta A_i = \sum ^m_{i=1}\sum ^n_{j=1} f\big (r^*_i\cos \theta , r^*_i\sin \theta \big )r^*_i \Delta r \Delta \theta \]

Writing \(\displaystyle {g(r,\theta ) = r f(r\cos \theta , r \sin \theta )}\) then \[\lim _{n,m\rightarrow \infty } \sum ^m_{i=1}\sum ^n_{j=1}g(r^*_i, \theta ^*_j)\hspace {0.1cm} \Delta r\hspace {0.1cm} \Delta \theta = \int ^{\beta }_{\alpha } \int ^b_a g(r,\theta )\hspace {0.1cm}dr\hspace {0.1cm}d\theta \]

\[\therefore \hspace {0.5cm} \iint \limits _R f(x,y)\hspace {0.1cm}dA = \int ^{\beta }_{\alpha } \int ^b_a f(r\cos \theta , r\sin \theta )\hspace {0.1cm} r\hspace {0.1cm} dr\hspace {0.1cm} d\theta \]

NB Replace \(dA\) by \(r\,dr\,d\theta \).

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