Textbook Question
Tangents and normals: Let a polar curve be described by r = f(θ), and let ℓ be the line tangent to the curve at the point P(x,y) = P(r,θ) (see figure).
b. Explain why tan θ = y/x.
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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 12.4.81b
Volume of a hyperbolic cap Consider the region R bounded by the right branch of the hyperbola x²/a² - y²/b² = 1 and the vertical line through the right focus.
b. What is the volume of the solid that is generated when R is revolved about the y-axis?
Verified step by step guidance1
Identify the region R bounded by the right branch of the hyperbola given by the equation \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) and the vertical line through the right focus. Recall that the foci of the hyperbola are located at \(x = \pm c\), where \(c = \sqrt{a^{2} + b^{2}}\). So the vertical line is \(x = c\).
Express \(x\) as a function of \(y\) from the hyperbola equation to describe the boundary curve. Solve for \(x\) to get \(x = a \sqrt{1 + \frac{y^{2}}{b^{2}}}\), which represents the right branch of the hyperbola.
Set up the volume integral for the solid generated by revolving the region R about the y-axis. Since the region is bounded between \(x = a \sqrt{1 + \frac{y^{2}}{b^{2}}}\) and \(x = c\), and \(y\) varies between appropriate limits, use the method of cylindrical shells or washers. Here, the washer method is convenient:
The volume element when revolving around the y-axis is given by \(dV = \pi (R_{outer}^{2} - R_{inner}^{2}) dy\), where \(R_{outer}\) and \(R_{inner}\) are the distances from the y-axis to the outer and inner boundaries of the region. In this case, \(R_{outer} = c\) and \(R_{inner} = a \sqrt{1 + \frac{y^{2}}{b^{2}}}\). Determine the limits of integration for \(y\) by finding the intersection points of the hyperbola and the vertical line \(x = c\).
Write the volume integral as \(V = \pi \int_{y_{min}}^{y_{max}} \left(c^{2} - a^{2} \left(1 + \frac{y^{2}}{b^{2}}\right) \right) dy\). Evaluate this integral over the determined limits to find the volume of the solid.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Equation and Properties of a Hyperbola
A hyperbola is defined by the equation x²/a² - y²/b² = 1, representing two branches. The right branch corresponds to x ≥ a. Understanding the shape and position of the hyperbola, including its foci located at (±c, 0) where c² = a² + b², is essential for identifying the region R and setting integration limits.
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Properties of Functions
Volume of Solids of Revolution
When a plane region is revolved around an axis, it generates a solid whose volume can be found using methods like the disk/washer or shell method. For revolution about the y-axis, the shell method is often convenient, integrating cylindrical shells formed by vertical slices of the region.
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04:48Finding Volume Using Disks
Setting Integration Limits Using the Focus and Boundary Lines
The region R is bounded by the hyperbola and the vertical line through the right focus (x = c). Correctly identifying this vertical boundary is crucial to determine the limits of integration for the volume calculation, ensuring the volume corresponds exactly to the specified region.
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08:01Integration Using Partial Fractions
Related Practice
Textbook Question
Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.
b. x = 2 + 5s, y = 1 + s and x = 4 + 10t, y = 3 + 2t
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Textbook Question
Channel flow Water flows in a shallow semicircular channel with inner and outer radii of 1 m and 2 m (see figure). At a point P(r,θ) in the channel, the flow is in the tangential direction (counterclock wise along circles), and it depends only on r, the distance from the center of the semicircles.
a. Express the region formed by the channel as a set in polar coordinates.
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. An object following the parametric curve x=2cos 2πt, y=2 sin 2πt circles the origin once every 1 time unit.
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Textbook Question
Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.
a. x = 1 + s, y = 2s and x = 1 + 2t, y = 3t
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. On every ellipse, there are exactly two points at which the curve has slope s, where s is any real number.