4.multiple - choice(5 points) medium what geometric shape approximates the area element when calculating…
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
Answer
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)