Textbook Question
49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.
y = 3
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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 12, Problem 12.1.82
81–88. Arc length Find the arc length of the following curves on the given interval.
x = 3 cos t, y = 3 sin t + 1; 0 ≤ t ≤ 2π
Verified step by step guidance1
Recall the formula for the arc length of a parametric curve given by \(x = x(t)\) and \(y = y(t)\) over the interval \(a \leq t \leq b\):
\[L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\]
Find the derivatives of \(x(t)\) and \(y(t)\) with respect to \(t\):
\[\frac{dx}{dt} = \frac{d}{dt}(3 \cos t) = -3 \sin t\]
\[\frac{dy}{dt} = \frac{d}{dt}(3 \sin t + 1) = 3 \cos t\]
Substitute these derivatives into the arc length formula under the square root:
\[\sqrt{(-3 \sin t)^2 + (3 \cos t)^2} = \sqrt{9 \sin^2 t + 9 \cos^2 t}\]
Use the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\) to simplify the expression inside the square root:
\[\sqrt{9 (\sin^2 t + \cos^2 t)} = \sqrt{9 \cdot 1} = 3\]
Set up the integral for the arc length over the interval \(0 \leq t \leq 2\pi\):
\[L = \int_0^{2\pi} 3 \, dt\]
This integral can then be evaluated to find the total arc length.
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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 as t. In this problem, x and y are given in terms of t, allowing the curve to be traced as t varies over the interval. Understanding parametric form is essential for applying calculus techniques to curves not defined explicitly as y = f(x).
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08:02
Parameterizing Equations
Arc Length Formula for Parametric Curves
The arc length of a curve defined parametrically by x(t) and y(t) from t = a to t = b is found using the integral ∫ from a to b of √[(dx/dt)² + (dy/dt)²] dt. This formula calculates the length by summing infinitesimal distances along the curve, requiring differentiation of both x and y with respect to t.
Recommended video:
06:29Arc Length of Parametric Curves
Differentiation of Trigonometric Functions
Calculating the derivatives dx/dt and dy/dt involves differentiating trigonometric functions like sine and cosine. Knowing that d/dt[cos t] = -sin t and d/dt[sin t] = cos t is crucial for correctly applying the arc length formula and simplifying the integral for evaluation.
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6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.
4x² - y² = 16
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Textbook Question
33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.
The region inside the circle r = 8 sin θ
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Textbook Question
45–60. Areas of regions Find the area of the following regions.
The region inside the outer loop but outside the inner loop of the limaçon r = 3 - 6 sin θ
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Textbook Question
15–30. Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.
x = cos t, y = sin² t; 0 ≤ t ≤ π
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Textbook Question
63–74. Arc length of polar curves Find the length of the following polar curves.
The complete circle r = a sin θ, where a > 0
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