Question

4.multiple - choice(5 points) medium what geometric shape approximates the area element when calculating areas in polar coordinates for a small angle $d\theta$? a a rectangle with height $r$ and width $d\theta$ b a triangle with base $r d\theta$ and height $r$ c a circular sector (wedge) with radius $r$ and angle $d\theta$ d a trapezoid formed between two radial lines

Explanation:

Step1: Recall polar - area formula concept

When calculating areas in polar coordinates for a small angle \(d\theta\), we consider the infinitesimal area element.

Step2: Analyze geometric shape

The area element \(dA\) in polar coordinates for a small angle \(d\theta\) and radius \(r\) is approximated by a circular sector (wedge) with radius \(r\) and central - angle \(d\theta\). The formula for the area of a circular sector with radius \(r\) and central - angle \(\alpha\) (in radians) is \(A=\frac{1}{2}r^{2}\alpha\). For an infinitesimal angle \(d\theta\), the area element \(dA = \frac{1}{2}r^{2}d\theta\).

Answer:

C. A circular sector (wedge) with radius \(r\) and angle \(d\theta\)