Find the exact length of the polar curve $r=2$ for $0\le \theta \le 2\pi$.

• A. $8\pi$
• B. $4\pi$
• C. $2\pi$
• D. $16$

Answer: (B)

Verified step by step guidance

1. Recognize that the given polar curve is \( r = 2 \), which represents a circle with radius 2 centered at the origin in polar coordinates.
2. Recall the formula for the length \( L \) of a polar curve \( r = f(\theta) \) from \( \theta = a \) to \( \theta = b \): \[ L = \int_{a}^{b} \sqrt{r(\theta)^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta \]
3. Since \( r = 2 \) is a constant, compute its derivative with respect to \( \theta \): \[ \frac{dr}{d\theta} = 0 \]
4. Substitute \( r = 2 \) and \( \frac{dr}{d\theta} = 0 \) into the length formula to simplify the integrand: \[ \sqrt{(2)^2 + 0^2} = \sqrt{4} = 2 \]
5. Set up the integral for \( \theta \) from 0 to \( 2\pi \): \[ L = \int_0^{2\pi} 2 \, d\theta \] Evaluate this integral to find the exact length of the curve.