Textbook Question
Finding Lengths of Polar Curves
Find the lengths of the curves in Exercises 21–28.
The curve r = cos³(θ/3), 0 ≤ θ ≤ π/4
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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 11.2.33
Surface Area
Find the areas of the surfaces generated by revolving the curves in Exercises 31-34 about the indicated axes.
x = t + √2, y = (t²/2) + √2t, −√2 ≤ t ≤ √2; y−axis
Verified step by step guidance1
Identify the parametric equations given: \(x = t + \sqrt{2}\) and \(y = \frac{t^{2}}{2} + \sqrt{2}t\), with the parameter \(t\) ranging from \(-\sqrt{2}\) to \(\sqrt{2}\).
Since the curve is revolved about the \(y\)-axis, use the formula for the surface area of a parametric curve revolved around the \(y\)-axis:
\[ S = \int_{a}^{b} 2\pi |x(t)| \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \]
Compute the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\):
- \(\frac{dx}{dt} = 1\)
- \(\frac{dy}{dt} = t + \sqrt{2}\)
Substitute \(x(t)\), \(\frac{dx}{dt}\), and \(\frac{dy}{dt}\) into the surface area integral:
\[ S = \int_{-\sqrt{2}}^{\sqrt{2}} 2\pi |t + \sqrt{2}| \sqrt{1^2 + (t + \sqrt{2})^2} \, dt \]
Evaluate the integral over the interval \([-\sqrt{2}, \sqrt{2}]\) to find the total surface area. Consider the absolute value in \(|t + \sqrt{2}|\) when setting up the integral, possibly splitting the integral if needed.
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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. Understanding how to work with x(t) and y(t) allows us to describe complex curves and is essential for setting up integrals in problems involving curves defined parametrically.
Recommended video:
Guided course
08:02
Parameterizing Equations
Surface Area of Revolution
The surface area generated by revolving a curve around an axis is found using an integral formula that involves the radius of revolution and the arc length element. For parametric curves, the formula integrates 2π times the radius times the differential arc length over the parameter interval.
Recommended video:
09:07Example 1: Minimizing Surface Area
Arc Length of Parametric Curves
Arc length for parametric curves is computed by integrating the square root of the sum of the squares of the derivatives of x and y with respect to the parameter. This measure is crucial for calculating surface areas and other quantities involving the length of the curve.
Recommended video:
06:29Arc Length of Parametric Curves
Related Practice
Textbook Question
Identifying Graphs
Match the parabolas in Exercises 1−4 with the following equations: x² = 2y, x² = −6y, y² = 8x, y² = −4x
Then find each parabola's focus and directrix.
Textbook Question
Ellipses and Eccentricity
Exercises 9–12 give the foci or vertices and the eccentricities of ellipses centered at the origin of the xy-plane. In each case, find the ellipse’s standard-form equation in Cartesian coordinates.
Vertices: (±10,0)
Eccentricity: 0.24
Textbook Question
Finding Polar Areas
Find the areas of the regions in Exercises 9–18.
Shared by the circles r = 1 and r = 2 sin θ
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Textbook Question
Cartesian to Polar Equations
Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.
(x + 2)² + (y − 5)² = 16"
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Textbook Question
Finding Cartesian from Parametric Equations
Exercises 1–18 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. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.
x=√(t+1), y=√t, t ≥ 0