Multiple Choice
Given the point with polar coordinates , which of the following polar coordinate pairs represents the same point?
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9. Polar Equations
Polar Coordinate System
Multiple Choice
Find the exact length of the polar curve for .
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Verified step by step guidance1
Recognize that the given polar curve is \( r = 2 \), which represents a circle with radius 2 centered at the origin in polar coordinates.
Recall the formula for the length \( L \) of a polar curve \( r = f(\theta) \) from \( \theta = a \) to \( \theta = b \):
\[ L = \int_{a}^{b} \sqrt{r(\theta)^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta \]
Since \( r = 2 \) is a constant, compute its derivative with respect to \( \theta \):
\[ \frac{dr}{d\theta} = 0 \]
Substitute \( r = 2 \) and \( \frac{dr}{d\theta} = 0 \) into the length formula to simplify the integrand:
\[ \sqrt{(2)^2 + 0^2} = \sqrt{4} = 2 \]
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.
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