Length in Polar Coordinates

Find the lengths of the curves given by the polar coordinate equations in Exercises 51–54.

r = √(1 + cos 2θ), −π/2 ≤ θ ≤ π/2

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Recall the formula for the length of a curve given in polar coordinates: the arc length \( L \) from \( \theta = a \) to \( \theta = b \) is given by \[ L = \int_{a}^{b} \sqrt{r(\theta)^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta \]

Identify the given function and interval: here, \( r(\theta) = \sqrt{1 + \cos 2\theta} \) and the interval is \( -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \).

Compute the derivative \( \frac{dr}{d\theta} \). Since \( r(\theta) = (1 + \cos 2\theta)^{1/2} \), use the chain rule: \[ \frac{dr}{d\theta} = \frac{1}{2}(1 + \cos 2\theta)^{-1/2} \cdot (-2 \sin 2\theta) = -\frac{\sin 2\theta}{\sqrt{1 + \cos 2\theta}} \]

Substitute \( r(\theta) \) and \( \frac{dr}{d\theta} \) into the arc length formula: \[ L = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{\left(\sqrt{1 + \cos 2\theta}\right)^2 + \left(-\frac{\sin 2\theta}{\sqrt{1 + \cos 2\theta}}\right)^2} \, d\theta \]

Simplify the expression inside the square root and then evaluate the integral over the given interval to find the length of the curve.

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주요 개념

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Arc Length in Polar Coordinates

The arc length of a curve defined in polar coordinates r(θ) from θ = a to θ = b is given by the integral ∫_a^b √(r(θ)^2 + (dr/dθ)^2) dθ. This formula accounts for both the radial distance and the rate of change of the radius with respect to the angle, allowing calculation of the curve's length.

Differentiation of Polar Functions

To find the derivative dr/dθ, you must differentiate the given polar function r(θ) with respect to θ. This step is essential for the arc length formula and often involves applying chain and trigonometric differentiation rules, especially when r is expressed with trigonometric functions like cos(2θ).

Trigonometric Identities and Simplification

Using trigonometric identities, such as double-angle formulas, can simplify expressions involving cos(2θ) or sin(2θ). Simplifying r(θ) and its derivative before integration makes the arc length integral more manageable and sometimes allows for exact evaluation or easier numerical approximation.

Length in Polar CoordinatesFind the lengths of the curves given by the polar coordinate equations in Exercises 51–54.r = √(1 + cos 2θ), −π/2 ≤ θ ≤ π/2