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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.62b
Chapter 7, Problem 7.3.62b

61–62. Points of intersection and area
b. Compute the area of the region described.


f(x) = sinh x, g(x) = tanh x; the region bounded by the graphs of f, g, and x = ln 3

Verified step by step guidance
1
Identify the region bounded by the curves \(f(x) = \sinh x\), \(g(x) = \tanh x\), and the vertical line \(x = \ln 3\). To find the area, we need to determine the interval over which to integrate. Since the problem specifies \(x = \ln 3\) as a boundary, find the other boundary by finding the point where \(f(x)\) and \(g(x)\) intersect for \(x < \ln 3\).
Set the two functions equal to find their points of intersection: \(\sinh x = \tanh x\). Solve this equation to find the \(x\)-values where the curves meet. These will serve as the limits of integration along with \(x = \ln 3\).
Determine which function is on top and which is on the bottom between the limits of integration. This is important because the area between two curves \(f(x)\) and \(g(x)\) from \(a\) to \(b\) is given by \(\int_a^b |f(x) - g(x)| \, dx\). Identify \(f(x)\) or \(g(x)\) as the upper function and the other as the lower function in this interval.
Set up the definite integral for the area: \(\text{Area} = \int_{a}^{\ln 3} \bigl| f(x) - g(x) \bigr| \, dx\), where \(a\) is the intersection point found in step 2. Since you know which function is on top, you can write the integral without absolute value as \(\int_{a}^{\ln 3} (\text{upper function} - \text{lower function}) \, dx\).
Evaluate the integral by integrating the difference of the functions. Recall the derivatives and integrals of hyperbolic functions: \(\frac{d}{dx} \sinh x = \cosh x\), \(\frac{d}{dx} \tanh x = \text{sech}^2 x\). Use these to find antiderivatives and then apply the Fundamental Theorem of Calculus to compute the definite integral.

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Key Concepts

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

Hyperbolic Functions

Hyperbolic functions like sinh(x) and tanh(x) are analogs of trigonometric functions but based on hyperbolas. sinh(x) = (e^x - e^{-x})/2 and tanh(x) = sinh(x)/cosh(x). Understanding their properties and graphs is essential to identify the region bounded by these curves.
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Asymptotes of Hyperbolas

Finding Points of Intersection

To determine the bounded region, it is crucial to find where the two functions intersect. This involves solving the equation sinh(x) = tanh(x) to find the x-values where the curves meet, which define the limits of integration for the area calculation.
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Critical Points

Definite Integral for Area Between Curves

The area between two curves f(x) and g(x) from a to b is found by integrating the difference |f(x) - g(x)| dx over [a, b]. Identifying which function is on top in the interval and using the given boundary x = ln(3) allows computation of the enclosed area.
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Finding Area Between Curves on a Given Interval
Related Practice
Textbook Question

Oil consumption Starting in 2018 (t=0), the rate at which oil is consumed by a small country increases at a rate of 1.5%/yr, starting with an initial rate of 1.2 million barrels/yr.


b. Find the function that gives the amount of oil consumed between t=0 and any future time t.

Textbook Question

Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.


b. Find the total energy (in MW-yr) used by the city over four full years beginning at t=0.

Textbook Question

37–38. Caffeine After an individual drinks a beverage containing caffeine, the amount of caffeine in the bloodstream can be modeled by an exponential decay function, with a half-life that depends on several factors, including age and body weight. For the sake of simplicity, assume the caffeine in the following drinks immediately enters the bloodstream upon consumption.


An individual consumes two cups of coffee, each containing 90 mg of caffeine, two hours apart. Assume the half-life of caffeine for this individual is 5.7 hours.


b. Determine the amount of caffeine in the bloodstream 1 hour after drinking the second cup of coffee.

Textbook Question

Theorem 7.8

Differentiate sinh⁻¹ x = ln (x + √(x² + 1)) to show that d/dx (sinh⁻¹ x) = 1 / √(x² + 1).

Textbook Question

Overtaking City A has a current population of 500,000 people and grows at a rate of 3%/yr. City B has a current population of 300,000 and grows at a rate of 5%/yr.

b. Suppose City C has a current population of y₀ < 500,000 and a growth rate of p > 3%/yr. What is the relationship between y₀ and p such that Cities A and C have the same population in 10 years?

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.


b. ln 0 = 1