Q. 3 - Calculus

Question

Let $\mathrm{\alpha }<\mathrm{\beta }$ and $\mathrm{a} < \mathrm{b}$. In polar coordinates, a polar rectangle is bounded by the two rays $\mathrm{\theta } = \mathrm{\alpha }$ and $\mathrm{\theta } = \mathrm{\beta }$ and the two circles $\mathrm{r} = \mathrm{a}$ and $\mathrm{r} = \mathrm{b}$. Sketch a polar rectangle and explain why this is the basic region for integration in the polar coordinate plane

Step-by-Step Solution

Answer

In the polar coordinate plane, the rectangle is the basic region for integration.

Step 1: Given information

Rectangle is bounded by the two rays $\mathrm{\theta }=\mathrm{\alpha } \mathrm{and} \mathrm{\theta }=\mathrm{\beta }$and the two circles $\mathrm{r}=\mathrm{a}$ and $\mathrm{r}=\mathrm{b}$

Step 2: Finding integration in the polar coordinates

The goal of this exercise is to draw a polar rectangle that is limited by two rays and two circles.

Ray of equations is $\mathrm{\theta }=\mathrm{\alpha } \mathrm{and} \mathrm{\theta }=\mathrm{\beta }$

Two circles of equations are $\mathrm{r}=\mathrm{a} \mathrm{and} \mathrm{r}=\mathrm{b}$

A polar rectangle A B C D is a common area surrounded by lines and circles.

The integration constraints in the common region are as follows:

$\mathrm{\theta }=\mathrm{\alpha } \mathrm{and} \mathrm{\theta }=\mathrm{\beta }$

$\mathrm{r}=\mathrm{a} \mathrm{and} \mathrm{r}=\mathrm{b}$

As a result, the area of rectangles A B C D is $\mathrm{A}={\int }_{\mathrm{\theta }-\mathrm{\alpha }}^{\mathrm{\theta }-\mathrm{\beta }} {\int }_{\mathrm{r}=\mathrm{a}}^{\mathrm{r}=\mathrm{b}} \mathrm{f}(\mathrm{r},\mathrm{\theta })\mathrm{rdrd\theta }$

The extent of the common region is defined here by lines and circles. As a result, on the polar coordinate plane, the rectangle is the basic region for integration.