Question

When we use rectangular coordinates to approximate the area of a region, we subdivide the region into vertical strips and approximate the area by using a sum of areas of rectangles. Explain why we use a wedge (i.e., a sector of a circle), and not a rectangle, when we employ polar coordinates to compute an area.

Short answer

Area in the polar system $=\text{Area of}{1}^{\text{st}}\text{sector}+\text{Area of}{2}^{\text{nd}}\text{sector}+\dots$

Step-by-step solution

Given information

Rectangular coordinates to approximate the area of a region,

Simplifications 

The goal of this topic is to explain why we compute an area in polar coordinates using a wedge rather than a rectangle.

The area of a region is partitioned into vertical strips in a rectangular coordinate system, and the area is approximated by adding the areas of rectangles.

The radius vector r describes the location of each point in the plane in the polar coordinate system. The length of the radius increases/decreases radially from the fixed point known as the pole O. With the help of an angle $\mathrm{\theta },$the position of the radius vector r is specified with reference to the fixed line known as the beginning line.

The region is believed to be developed using numerous sectors in order to find an area in the polar system. This is $\text{Area in polar system}=\text{Area of}{1}^{\mathrm{st}}\text{sector}+\text{Area of}{2}^{\text{nd}}\text{sector}+\dots$