Q. 3. - Calculus

Question

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

Step-by-Step Solution

Answer

The region's area is approximated using "wedges," which refers to a circle's sector.

A wedge-shaped slice of the region is determined by a minor change in $\Delta \theta$

Step 1: Given information

Consider a function $r=f\left(\theta \right)$ in polar coordinates. Where $\theta$ is angular rotation 

Step 2: Explain why we use a “wedge” and not a rectangle when we use polar coordinates to compute an area.

A circle's area is calculated by splitting it into an infinite number of wedges produced by radii drawn from the center. When these wedges are rearranged, they form a rectangle with a height equal to the circle's radius and a base length equal to half of the circle's diameter.

The region's area is approximated using "wedges," which refers to a circle's sector.

A wedge-shaped slice of the region is determined by a minor change in $\Delta \theta$