Question

What is the area of a loop of the curve \( r = a \sin 30^\circ \)?

• A. \( \frac{\pi a^2}{6} \)
• B. \( \frac{\pi a^2}{8} \)
• C. \( \frac{\pi a^2}{12} \)
• D. \( \frac{\pi a^2}{24} \)

Hint:

For polar curves, use the formula for the area of a loop, which involves integrating \( r^2 \) over the limits of the loop.

Answer 1

The Correct Option is D

Step 1: Area under the curve. The area of a loop of a polar curve is given by the formula \( A = \frac{1}{2} \int_{\theta_1}^{\theta_2} r^2 d\theta \).
Step 2: Conclusion. Thus, the area of the loop is \( \frac{\pi a^2}{24} \).