Q. 7 - Calculus

Question

Let $0<f_1\left(\theta \right)<f_2\left(\theta \right)$ on the interval $[\alpha ,\beta ]$. What does the integral ${\int }_{\alpha }^{\beta }{\int }_{f_1\left(\theta \right)}^{f_2\left(\theta \right)}drd\theta$ represent in a rectangular $\theta r$  coordinate system? What does the integral represent in a polar coordinate system?

Step-by-Step Solution

Answer

In the $\theta r$-coordinate system, the integral represents the area of a rectangle bounded by the lines $r=f_1\left(\theta \right),r=f_2\left(\theta \right),\theta =\alpha$ and $\theta =\beta$.

In a polar coordinate system, the integral represents the volume of the solid region between the circles $r=f_1\left(\theta \right)$ and $r=f_2\left(\theta \right)$ bounded by the rays $\theta =\alpha$ and $\theta =\beta$

Step 1 Given Information

Consider the inequality $0<f_1\left(\theta \right)<f_2\left(\theta \right)$ on the closed interval $[\alpha ,\beta ]$ and the integral ${\int }_a^{A{f}_2\left(\theta \right)}{\int }_{{\delta }_1\left(\theta \right)}drd\theta$
The objective is to explain the meaning of integral in the rectangular $\theta r$-coordinate system and in the polar coordinate system.

Step 2 Calculation

Consider the rectangular $\theta r$-coordinate system.
The region is bounded by the following equations:
$r=f_1\left(\theta \right)$

$r=f_2\left(\theta \right)$

$\theta =\alpha$

$\theta =\beta$

First two curves are horizontal lines in $\theta r$-coordinate system and the last two curves are the vertical lines.
Therefore, in the $\theta r$-coordinate system, the integral represents the area of a rectangle bounded by the lines $r=f_1\left(\theta \right),r=f_2\left(\theta \right),\theta =\alpha$ and $\theta =\beta$.

Consider a polar coordinate system.
The region is bounded by the following equations:
$r=f_1\left(\theta \right)$

$r=f_2\left(\theta \right)$

$\theta =\alpha$

$\theta =\beta$

The first two curves are the circles in a polar coordinate system and last two are the radial rays originating from the origin.

Therefore, in a polar coordinate system, the integral represents the volume of the solid region between the circles $r=f_1\left(\theta \right)$ and $r=f_2\left(\theta \right)$ bounded by the rays $\theta =\alpha$ and $\theta =\beta$