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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblem 11.PE.54
Chapter 11, Problem 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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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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
Related Practice
Textbook Question

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

Textbook Question

Graphing Conic Sections


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


y² = −(8/3)x

Textbook Question

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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Textbook Question

Finding Parametric Equations


Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².


a. once clockwise.


(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)

Textbook Question

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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Textbook Question

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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