Textbook Question
31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.
(-4, 4√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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Guided course
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
49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.
y = 3
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
37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.
The line that passes through the points P(1, 1) and Q(3, 5), oriented in the direction of increasing x
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Textbook Question
81–88. Arc length Find the arc length of the following curves on the given interval.
x = eᵗ sin t, y = eᵗ cos t; 0 ≤ t ≤ 2π
Textbook Question
37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.
r = 6 cos θ + 8 sin θ