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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.PE.54
Capitolo 11, Problema 11.PE.54

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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1
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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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.
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Intro to Polar Coordinates

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).
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Intro to Polar Coordinates

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.
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Verifying Trig Equations as Identities
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Identifying Parametric Equations in the Plane


Exercises 1–6 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle.


x = √t, y = 1 − √t, t ≥ 0

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Graphing Conic Sections


Sketch the parabolas in Exercises 55–58. Include the focus and directrix in each sketch.


y² = −(8/3)x

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Lengths of Curves


Find the lengths of the curves in Exercises 13–19.


x = 5 cos t − cos 5t, y = 5 sin t − sin 5t, 0 ≤ t ≤ π/2

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Cartesian to Polar Equations


Find polar equations for the circles in Exercises 33–36. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations.


x² + y² + 5y = 0

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Area in Polar Coordinates


Find the areas of the regions in the polar coordinate plane described in Exercises 47–50.


Inside the cardioid r = 2(1 + sin θ) and outside the circle r = 2 sin θ

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Cycloid


a. Find the length of one arch of the cycloid x = a(t − sin t), y = a(1 − cos t).

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