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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.4.81a
Chapter 12, Problem 12.4.81a

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.
a. What is the volume of the solid that is generated when R is revolved about the x-axis?

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1
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 \(y\) as a function of \(x\) from the hyperbola equation. Solve for \(y\) to get \(y = \pm \frac{b}{a} \sqrt{x^{2} - a^{2}}\). Since the region is bounded by the right branch, consider \(x \geq a\) and the positive \(y\) values for the upper half (or use symmetry if needed).
Set up the volume integral using the method of disks or washers for revolution about the x-axis. The volume element is \(dV = \pi y^{2} dx\), so the volume \(V\) is given by the integral \(V = \pi \int_{a}^{c} y^{2} \, dx\).
Substitute \(y^{2}\) from the expression found: \(y^{2} = \left( \frac{b}{a} \right)^{2} (x^{2} - a^{2})\). Thus, the integral becomes \(V = \pi \int_{a}^{c} \left( \frac{b^{2}}{a^{2}} (x^{2} - a^{2}) \right) dx\).
Evaluate the integral by integrating term-by-term: \(\int_{a}^{c} (x^{2} - a^{2}) dx = \int_{a}^{c} x^{2} dx - a^{2} \int_{a}^{c} dx\). After integration, multiply by \(\pi \frac{b^{2}}{a^{2}}\) to find the volume expression.

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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 to identify the region R and set integration limits.
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Properties of Functions

Volume of Solids of Revolution

When a plane region is revolved around an axis, the volume of the resulting solid can be found using methods like the disk/washer or shell method. For revolution about the x-axis, the disk method integrates π[y(x)]² dx over the interval, where y(x) is the function describing the boundary curve.
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Finding 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 at x = c. Correctly identifying this vertical boundary determines the integration limits for the volume calculation. This step ensures the volume corresponds exactly to the specified hyperbolic cap.
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Integration Using Partial Fractions
Related Practice
Textbook Question

11–14. Working with parametric equations Consider the following parametric equations.

a. Make a brief table of values of t, x, and y.

b. Plot the (x, y) pairs in the table and the complete parametric curve, indicating the positive orientation (the direction of increasing t).


x=−t+6, y=3t−3; −5≤t≤5 

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Textbook Question

11–14. Working with parametric equations Consider the following parametric equations.

a. Make a brief table of values of t, x, and y.

b. Plot the (x, y) pairs in the table and the complete parametric curve, indicating the positive orientation (the direction of increasing t).


x=2 t,y=3t−4;−10≤d≤10 

Textbook Question

The ellipse and the parabola: Let R be the region bounded by the upper half of the ellipse x²/2 + y² = 1 and the parabola y = x²/√2

a. Find the area of R

Textbook Question

67–72. Derivatives Consider the following parametric curves.

a. Determine dy/dx in terms of t and evaluate it at the given value of t.


x = cos t, y = 8 sin t; t = π/2

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Textbook Question

67–72. Derivatives Consider the following parametric curves.

a. Determine dy/dx in terms of t and evaluate it at the given value of t.


x = t + 1/t, y = t − 1/t; t = 1

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Textbook Question

(Use of Tech) Finger curves: r = f(θ) = cos(aᶿ) - 1.5, where a = (1 + 12π)^(1/(2π)) ≈ 1.78933

a. Show that f(0) = f(2π) and find the point on the curve that corresponds to θ = 0 and θ = 2π.

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