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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?

Verified step by step guidance
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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Intro to Polar Coordinates

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
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

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

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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.

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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

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

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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