Choosing the area element of Polar double integration.

Question

How do I divide the radius into subsections?

In one situation the region can be inside two circles and $\Delta r$ might equal for instance $\dfrac{2 - 1}{n}$. With another two radii that keep the variable in that format, like $\Delta r = \dfrac{cos(2\theta) - cos(\theta)}{n}$ how do you calculate the $\Delta r$ that's uniform everywhere? If I pick a sample of $\theta$ then I get uniform sub regions yet that seems like it should change the boundaries of integration. If the value of $\theta$ would vary then the regions don't have equal lengths of radius in the approximation. How do I always see this un evaluated expression as sub regions in the approximation walkthroughs?

Answer 1

We first choose a differential area of $dr$ x $ rd\theta$ (highlighted in orange color), and separate the integrations along two different directions:

1. Radial direction by integrate dr from $r_1$ to $r_2$: $\int \limits_{ r_1}^{ r_2}$dr

2. Circumferential direction by integrating d$\theta$ from $\theta_1$ to $\theta_2$: $\int \limits_{\theta _1}^{\theta_2}d\theta$

\begin{align}\int \limits_{ r_1}^{ r_2}rdr \int \limits_{\theta _1}^{\theta_2} d\theta \end{align}

In this process, we do not include $\theta$ in expression of dr , the variation of $\theta$ is never in the picture of integration along the radial direction.

Hope this helps.