The equation of a closed curve in two-dimensional polar coordinates is given by
r =
2
√𝜋 (1 - sin 𝜃) . The area enclosed by the curve is ________ (answer in
integer).
Official Answer Key:
6.0
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.42)**
**1. Intuitive Physical Setup & Given Parameters**:
A closed curve in 2D polar coordinates is given by:
$r(\theta) = \frac{2}{\sqrt{\pi}}(1 - \sin\theta) \quad \text{for } 0 \le \theta \le 2\pi$
(a cardioid curve oriented along the $y$-axis).
We are asked to find the **area enclosed by the curve** (answer in integer).
**2. Step-by-Step Calculus Derivation**:
1. **Area Enclosed by a Polar Curve Formula**:
$A = \frac{1}{2} \int_0^{2\pi} r^2(\theta) d\theta$
2. **Substitute $r(\theta)$**:
$r^2(\theta) = \left[\frac{2}{\sqrt{\pi}}(1 - \sin\theta)\right]^2 = \frac{4}{\pi} (1 - \sin\theta)^2 = \frac{4}{\pi} (1 - 2\sin\theta + \sin^2\theta)$
3. **Set Up the Integral**:
$A = \frac{1}{2} \times \frac{4}{\pi} \int_0^{2\pi} (1 - 2\sin\theta + \sin^2\theta) d\theta = \frac{2}{\pi} \int_0^{2\pi} (1 - 2\sin\theta + \sin^2\theta) d\theta$
4. **Evaluate Each Integral Over $[0, 2\pi]$**:
- $\int_0^{2\pi} 1 d\theta = 2\pi$
- $\int_0^{2\pi} \sin\theta d\theta = 0$
- $\int_0^{2\pi} \sin^2\theta d\theta = \int_0^{2\pi} \frac{1 - \cos 2\theta}{2} d\theta = \frac{2\pi}{2} = \pi$
5. **Sum the Contributions**:
$\int_0^{2\pi} (1 - 2\sin\theta + \sin^2\theta) d\theta = 2\pi - 0 + \pi = 3\pi$
6. **Compute Enclosed Area $A$**:
$A = \frac{2}{\pi} \times 3\pi = 6$
**3. Comprehensive Options & Distractor Analysis**:
- **Evaluation Criteria**: Numerical Answer Type (NAT).
- **Exact Analytical Result**: $A = 6$.
- **Official Range**: Integer answer: **6**.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **6.0** (Acceptable Range: **6 to 6**)
- **Exam Takeaway**: For a cardioid $r = a(1 - \sin\theta)$, the enclosed area is $A = \frac{3}{2}\pi a^2$. Here $a = \frac{2}{\sqrt{\pi}}$, giving $A = \frac{3}{2}\pi \left(\frac{4}{\pi}\right) = 6$.
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