Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Problem 7.3.69

Chapter 7, Problem 7.3.69

Catenary arch The portion of the curve y =17/15 - cosh x that lies above the x-axis forms a catenary arch. Find the average height of the arch above the x-axis.

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Identify the function describing the arch: $y = \frac{17}{15} - \cosh x$. The arch is the portion of this curve where $y > 0$ (above the x-axis).

Find the points where the arch intersects the x-axis by solving $\frac{17}{15} - \cosh x = 0$. This will give the interval $[a, b]$ over which the arch lies above the x-axis.

Set up the formula for the average height of the arch over the interval $[a, b]$: $\text{Average height} = \frac{1}{b - a} \int_a^b \left( \frac{17}{15} - \cosh x \right) \, dx$.

Evaluate the integral $\int_a^b \left( \frac{17}{15} - \cosh x \right) \, dx$ by integrating each term separately: $\int_a^b \frac{17}{15} \, dx$ and $\int_a^b \cosh x \, dx$.

Substitute the limits $a$ and $b$ into the evaluated integral, then divide by $(b - a)$ to find the average height of the arch above the x-axis.

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Key Concepts

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

Catenary Curve

A catenary curve is the shape formed by a flexible chain or cable hanging under its own weight when supported at its ends. It is described by the hyperbolic cosine function, y = a cosh(x/a), and differs from a parabola. Understanding the properties of cosh(x) is essential to analyze the given curve.

Average Value of a Function

The average value of a continuous function f(x) over an interval [a, b] is given by (1/(b - a)) times the integral of f(x) from a to b. This concept helps find the mean height of the arch by integrating the curve's height over the domain where it lies above the x-axis.

Definite Integration and Area Under Curve

Definite integration calculates the exact area under a curve between two points. For the catenary arch, integrating y = 17/15 - cosh(x) over the interval where y ≥ 0 provides the total height accumulation, which is then used to find the average height.

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