Question

a. Sketch the graphs of the functions f and g and find the \(x\) -coordinate of the points at which they intersect. b. Compute the area of the region described. \(f(x)=\operatorname{sech} x, g(x)=\tanh x ;\) the region bounded by the graphs of \(f, g,\) and the \(y\) -axis

Short answer

The approximate area of the region is 0.468 square units.

Step-by-step solution

Identify the points of intersection and sketch the graphs of the functions

To find the points of intersection between the given functions, f(x) and g(x), we need to set f(x) = g(x): $${\operatorname{sech}} x = \tanh x$$ Now, recall that: $${\operatorname{sech}} x = \frac{1}{\cosh x} \text{ and } \tanh x = \frac{\sinh x}{\cosh x}$$ Setting these two equal to each other and solving for x yields: $$\frac{1}{\cosh x} = \frac{\sinh x}{\cosh x}$$ Since the denominators of the fractions are equal, setting the numerators equal to each other: $$1 = \sinh x$$ The solution to this equation, x, gives the x-coordinates of the points of intersection between f(x) and g(x). Solving for x, we find: $$x = \sinh^{-1}(1)$$ Use a calculator to find the numerical approximation: $$x \approx 0.881$$ The graphs of the functions intersect at x ≈ 0.881. Now sketch the graphs of the functions on the same axes. Note that both functions will intersect the y-axis at x = 0.

Set up the integrals to compute the area

To compute the area of the region described, we need to find the difference between the integrals of the functions over the interval from the y-axis to the point of intersection. The expression for the area (A) is: $$A = \int_{0}^{0.881} (\operatorname{sech} x - \tanh x) \, dx$$

Evaluate the integral

Now, it's time to evaluate the integral using proper techniques or a calculator: $$A \approx \int_{0}^{0.881} (\operatorname{sech} x - \tanh x) \, dx \approx 0.468$$

State the final answer

From the calculations, we find that the area of the region bounded by the graphs of f(x), g(x), and the y-axis is approximately 0.468 square units.

Key concepts

Understanding Hyperbolic Functions

Hyperbolic functions are similar to trigonometric functions but are defined using the exponential function. They are useful in many areas of calculus and appear often in calculations involving hyperbolas. Here are a couple of key hyperbolic functions you'll encounter:

• Hyperbolic Secant (\( \operatorname{sech} x \)): Defined as \( \operatorname{sech} x = \frac{1}{\cosh x} \), where \( \cosh x = \frac{e^x + e^{-x}}{2} \). This function decreases as x moves away from zero.
• Hyperbolic Tangent (\( \tanh x \)): Given by \( \tanh x = \frac{\sinh x}{\cosh x} \), where \( \sinh x = \frac{e^x - e^{-x}}{2} \). The \( \tanh x \) function ranges between -1 and 1, providing a smooth transition from negative to positive values through zero.

These hyperbolic functions behave similarly to their trigonometric counterparts and are particularly important in areas such as calculus, physics, and engineering. When solving problems involving hyperbolic functions, it is often necessary to work with their identities and derivatives.

Integration Techniques

Integration is a fundamental concept in calculus, used to find areas under curves, among other applications. When integrating functions, the main aim is to determine the antiderivative or the indefinite integral of the function. Here are some common techniques to perform integration:

• Substitution: Useful when the integral contains a composition of functions. By substituting parts of the function, it becomes easier to integrate.
• Integration by Parts: Helpful for integrals of products of functions. Based on the product rule for differentiation, it systematically transforms difficult integrals into more manageable ones.
• Partial Fraction Decomposition: Applied when dealing with rational functions, where the integrand is expressed as a sum of simpler fractions which can be more easily integrated.

For the given problem, we calculated the area between two functions by setting up an integral over a specific interval. Evaluating this integral helped us find the precise area under the curve, representing the space enclosed by the two functions.

Area Under Curves and Bounded Regions

Finding the area under a curve is a key task in calculus. It involves calculating how much space is enclosed by the curve, a specific axis, or other bounding curves. Here's how you often perform these calculations:

• Identify the Region: First, determine the boundaries of the region involved. This may involve finding intersections of two functions or recognizing specific limits on the axes.
• Setting Up the Integral: Once the region is identified, set up an integral where the limits of integration are defined by the boundaries of the region. The integrand will be the function or the difference between two functions (for areas between curves).
• Evaluate the Integral: Solve the integral to find the area. This can require various techniques including numerical integration methods when the integrals are complex.

In the original exercise, by calculating the area between \( \operatorname{sech} x \) and \( \tanh x \), we determined the enclosed region from the y-axis to the intersection point. Understanding this process is crucial in many practical applications, such as physics and engineering, where determining physical volumes and areas is necessary.