# Choosing the area element of Polar double integration.

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?

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
1. 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.