91Ó°ÊÓ

Consider the inequality $0<f_1\left(\mathrm{θ}\right)<f_2\left(\mathrm{θ}\right)$on the closed interval $[\mathrm{Î\pm },\mathrm{β}]$and the integral ${∫}_a^{A{f}_2\left(\mathrm{θ}\right)}{∫}_{{\mathrm{δ}}_1\left(\mathrm{θ}\right)}drd\mathrm{θ}$
The objective is to explain the meaning of integral in the rectangular $\mathrm{θ}r$-coordinate system and in the polar coordinate system.

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

$r=f_2\left(\mathrm{θ}\right)$

$\mathrm{θ}=\mathrm{Î\pm }$

$\mathrm{θ}=\mathrm{β}$

First two curves are horizontal lines in $\mathrm{θ}r$-coordinate system and the last two curves are the vertical lines.
Therefore, in the $\mathrm{θ}r$-coordinate system, the integral represents the area of a rectangle bounded by the lines $r=f_1\left(\mathrm{θ}\right),r=f_2\left(\mathrm{θ}\right),\mathrm{θ}=\mathrm{Î\pm }$and $\mathrm{θ}=\mathrm{β}$.

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

$r=f_2\left(\mathrm{θ}\right)$

$\mathrm{θ}=\mathrm{Î\pm }$

$\mathrm{θ}=\mathrm{β}$

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 $r=f_1\left(\mathrm{θ}\right)$and $r=f_2\left(\mathrm{θ}\right)$bounded by the rays $\mathrm{θ}=\mathrm{Î\pm }$ and $\mathrm{θ}=\mathrm{β}$