Textbook Question
Find the area of the region bounded by the astroid x = cos³ t, y = sin³ t, for 0 ≤ t ≤ 2π
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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 12.4.43
39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin.
An ellipse with vertices (±5, 0), passing through the point (4, 3/5)
Verified step by step guidance1
Identify the standard form of the ellipse equation centered at the origin with a horizontal major axis: \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\).
From the vertices given as \((\pm 5, 0)\), determine the value of \(a\) since the vertices lie along the major axis. Here, \(a = 5\), so \(a^{2} = 25\).
Substitute \(a^{2} = 25\) into the ellipse equation to get \(\frac{x^{2}}{25} + \frac{y^{2}}{b^{2}} = 1\).
Use the point \((4, \frac{3}{5})\) that lies on the ellipse to substitute \(x = 4\) and \(y = \frac{3}{5}\) into the equation, resulting in \(\frac{4^{2}}{25} + \frac{(\frac{3}{5})^{2}}{b^{2}} = 1\).
Solve the resulting equation for \(b^{2}\) to find the value of \(b^{2}\), which completes the equation of the ellipse.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Form of an Ellipse Equation
An ellipse centered at the origin with a horizontal major axis has the equation (x²/a²) + (y²/b²) = 1, where 'a' is the distance from the center to each vertex along the x-axis, and 'b' is the distance along the y-axis. Knowing the vertices helps determine 'a', which is essential for writing the ellipse equation.
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Parameterizing Equations of Circles & Ellipses
Using a Point on the Ellipse to Find Parameters
Substituting a known point on the ellipse into the standard equation allows solving for the unknown parameter 'b'. This step ensures the ellipse passes through the given point, refining the equation to fit the specific ellipse described.
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Relationship Between 'a', 'b', and the Shape of the Ellipse
The values of 'a' and 'b' determine the ellipse's shape and size. 'a' is the semi-major axis length, and 'b' is the semi-minor axis length. Understanding their relationship helps in interpreting the ellipse's geometry and verifying the correctness of the equation.
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Related Practice
Textbook Question
{Use of Tech} Implicit function graph Explain and carry out a method for graphing the curve x = 1 + cos² y − sin² y using parametric equations and a graphing utility.
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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.
x² = 12y
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Textbook Question
53–56. Simple curves Tabulate and plot enough points to sketch a graph of the following equations.
r = 1 - cos θ
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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.
A circle centered at the origin with radius 4, generated counterclockwise
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Textbook Question
63–74. Arc length of polar curves Find the length of the following polar curves.
The complete cardioid r = 4 + 4 sin θ
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