Textbook Question
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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Ch. 11 - Parametric Equations and Polar Coordinates
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 11, Problem 11.PE.17
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
Verified step by step guidance1
Recall the formula for the length of a parametric curve given by \(x = x(t)\) and \(y = y(t)\) over the interval \(a \leq t \leq b\) is:
\[L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\]
Identify the given functions:
\[x(t) = 5 \cos t - \cos 5t\]
\[y(t) = 5 \sin t - \sin 5t\]
and the interval is \(0 \leq t \leq \frac{\pi}{2}\).
Compute the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\):
For \(x(t)\), use the derivative of cosine:
\[\frac{dx}{dt} = -5 \sin t + 5 \sin 5t\]
For \(y(t)\), use the derivative of sine:
\[\frac{dy}{dt} = 5 \cos t - 5 \cos 5t\]
Substitute these derivatives into the arc length formula under the square root:
\[L = \int_0^{\frac{\pi}{2}} \sqrt{\left(-5 \sin t + 5 \sin 5t\right)^2 + \left(5 \cos t - 5 \cos 5t\right)^2} \, dt\]
Simplify the expression inside the square root if possible, then set up the integral for evaluation. This integral may require numerical methods or further trigonometric simplifications to solve.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Parametric Equations
Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves. Understanding how to work with parametric forms is essential for calculating properties like curve length.
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Guided course
08:02
Parameterizing Equations
Arc Length Formula for Parametric Curves
The length of a curve defined parametrically by x(t) and y(t) from t = a to t = b is found by integrating the square root of the sum of the squares of the derivatives: ∫ₐᵇ √[(dx/dt)² + (dy/dt)²] dt. This formula generalizes the Pythagorean theorem to infinitesimal segments along the curve.
Recommended video:
06:29Arc Length of Parametric Curves
Differentiation of Trigonometric Functions
Calculating the derivatives dx/dt and dy/dt requires knowledge of differentiating trigonometric functions like sine and cosine. Recognizing the derivatives (d/dt) sin t = cos t and (d/dt) cos t = -sin t is crucial for correctly applying the arc length formula to the given parametric equations.
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6:04Introduction to Trigonometric Functions
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
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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Textbook Question
Polar to Cartesian Equations
Sketch the lines in Exercises 23-28. Also, find a Cartesian equation for each line.
r cos (θ − 3π/4) = (√2)/2
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