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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 12.4.80a
Chapter 12, Problem 12.4.80a

Area of a sector of a hyperbola: 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 area of R?

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1
Identify the given hyperbola equation: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). The right branch corresponds to \(x \geq a\) since the hyperbola opens along the x-axis.
Recall that the foci of the hyperbola are located at \(x = \pm c\), where \(c = \sqrt{a^2 + b^2}\). The right focus is at \(x = c\).
Set up the integral for the area \(R\) bounded by the hyperbola and the vertical line through the right focus \(x = c\). The area can be expressed as an integral with respect to \(x\) from \(x = a\) to \(x = c\):
\[\text{Area} = \int_a^c y(x) \, dx,\] where \(y(x)\) is the positive branch of the hyperbola.
Solve for \(y\) from the hyperbola equation: \(y = \frac{b}{a} \sqrt{x^2 - a^2}\). Substitute this into the integral to get:
\[\text{Area} = \int_a^c \frac{b}{a} \sqrt{x^2 - a^2} \, dx.\]
Evaluate the integral using an appropriate substitution, such as \(x = a \sec \theta\), to simplify the square root and compute the definite integral from \(x = a\) to \(x = c\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Equation and Geometry of a Hyperbola

A hyperbola is defined by the equation x²/a² - y²/b² = 1, representing two symmetric branches. The right branch corresponds to x ≥ a. Understanding the shape and position of the hyperbola, including its vertices and foci, is essential to identify the region bounded by the curve and the vertical line through the focus.
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Asymptotes of Hyperbolas

Coordinates of the Focus of a Hyperbola

The foci of the hyperbola x²/a² - y²/b² = 1 lie at (±c, 0), where c = √(a² + b²). The vertical line through the right focus is x = c. Knowing the focus location helps define the boundary of the region R and sets the limits for integration when calculating the area.
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Area Calculation Using Definite Integrals

The area of region R can be found by integrating the function defining the hyperbola's upper branch between x = a and x = c. This involves setting up the integral of y = (b/a)√(x² - a²) with respect to x, then evaluating it to find the exact area bounded by the curve and the vertical line.
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Definition of the Definite Integral
Related Practice
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