Skip to main content
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?

Verified step by step guidance
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6m
Was this helpful?

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.
Recommended video:
06:21
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.
Recommended video:
04:48
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.
Recommended video:
08:01
Integration Using Partial Fractions
Related Practice
Textbook Question

Navigating A plane is 150 miles north of a radar station, and 30 minutes later it is 60 degree east of north at a distance of 100 miles from the radar station. Assume the plane flies on a straight line and maintains constant altitude during this 30-minute period.

a. Find the distance traveled during this 30-minute period.

22
views
Textbook Question

Regions bounded by a spiral: Let Rₙ be the region bounded by the nth turn and the (n+1)st turn of the spiral r = e⁻ᶿ in the first and second quadrants, for θ ≥ 0 (see figure).

a. Find the area Aₙ of Rₙ.


76
views
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.

52
views
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.

35
views
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

63
views
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 = 2 + 4t, y = 4 − 8t; t = 2

50
views