Given the polar curve $r=4+3\cdot \sin \theta$, what is the area enclosed by one complete loop of the curve?

• A. $21\pi$
• B. $\frac{25\pi }{2}$
• C. $14\pi$
• D. $\frac{49\pi }{2}$

Answer: (D)

Verified step by step guidance

1. Recall that the area enclosed by a polar curve \( r = f(\theta) \) from \( \theta = a \) to \( \theta = b \) is given by the integral formula: \[ \text{Area} = \frac{1}{2} \int_a^b r^2 \, d\theta \]
2. Identify the given polar curve: \[ r = 4 + 3 \sin(\theta) \]
3. Determine the interval \( [a, b] \) for one complete loop of the curve. Since \( r = 4 + 3 \sin(\theta) \) is a sinusoidal function with period \( 2\pi \), one complete loop corresponds to \( \theta \) going from 0 to \( 2\pi \).
4. Set up the integral for the area using the formula: \[ \text{Area} = \frac{1}{2} \int_0^{2\pi} (4 + 3 \sin(\theta))^2 \, d\theta \]
5. Expand the square inside the integral and simplify the integrand: \[ (4 + 3 \sin(\theta))^2 = 16 + 24 \sin(\theta) + 9 \sin^2(\theta) \] Then, integrate each term separately over \( 0 \) to \( 2\pi \) using known integral formulas for \( \sin(\theta) \) and \( \sin^2(\theta) \).