Problem 43 - Calculus Early Transcendentals

Question

a. Use a graphing utility to sketch the graph of \(y=\operatorname{coth} x\), and then explain why \(\int_{5}^{10} \operatorname{coth} x d x \approx 5\). b. Evaluate \(\int_{5}^{10} \operatorname{coth} x d x\) analytically and use a calculator to arrive at a decimal approximation to the answer. How large is the error in the approximation in part (a)?

Step-by-Step Solution

Answer

Question: Approximate the integral of the hyperbolic cotangent function (coth(x)) from 5 to 10, and then find the analytical value and error in the approximation. Answer: The approximate value of the integral of coth(x) from 5 to 10 is about 5. The analytical value of the integral is approximately 4.04352, and the error in the approximation is about 0.95648.

Step 1: Sketch the graph of y=coth(x)

First, use a graphing utility (such as a graphing calculator or a graphing software like Desmos) to sketch the graph of y=coth(x). You will see that the graph of the coth(x) function looks like a smooth curve with a horizontal asymptote at y=0.

Step 2: Understand why the integral from 5 to 10 is approximately 5

Given that coth(x) has a horizontal asymptote at y=0 and approaches 1 as x approaches infinity, when x is between 5 and 10, coth(x) is close to 1. Therefore, the area under the curve between these limits is approximately 5 (as the width of the interval is also 5).

Step 3: Evaluate the integral analytically

We should recognize that the coth(x) function has fundamental properties, such as: \(\operatorname{coth} x = \frac{e^x + e^{-x}}{e^x - e^{-x}}\). To find the integral, we need its derivative. We can compute the derivative in the following way: \(\frac{d}{dx}\operatorname{coth}(x)=-\operatorname{csch}^2(x)\) Now we know the antiderivative, which is the hyperbolic cosecant function squared. To find the integral, we need to find the difference between the antiderivative evaluated at 10 and at 5: \(\int_{5}^{10} \operatorname{coth}(x) dx= -\operatorname{csch}(10)^2 + \operatorname{csch}(5)^2\).

Step 4: Compute the decimal approximation and error of \(\int_{5}^{10} \operatorname{coth} x d x\) analytically

Use a calculator to compute the numerical values of the expression: Approximate integral value = \(-\operatorname{csch}(10)^2 + \operatorname{csch}(5)^2\approx 4.04352 \) Now find the error in the approximation from part (a): Error = Approximate value from part (a) - Analytical value = \(5 - 4.04352\approx 0.95648\) So, the error in the approximation from part (a) is about 0.95648 units.

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but are based on hyperbolas instead of circles. One of the most widely used hyperbolic functions is the hyperbolic cotangent, written as \( \operatorname{coth} x \). This function is defined by the formula:

• \( \operatorname{coth} x = \frac{e^x + e^{-x}}{e^x - e^{-x}} \)

Hyperbolic functions arise in various areas of calculus, physics, and engineering, notably in dealing with problems involving hyperbolic geometry and certain differential equations.
A peculiar feature of \( \operatorname{coth} x \) is its horizontal asymptote at \( y = 0 \) as \( x \to \infty \), meaning the function levels off and approaches zero.
Additionally, its shape and values give insights into the structure of the area under the curve when analyzing integrals. Understanding these underpinnings can help us calculate integrals involving \( \operatorname{coth} x \) more intuitively.

Graphical Analysis

Graphical analysis involves using visual representations to understand and solve mathematical problems. For hyperbolic functions like \( y = \operatorname{coth} x \), graphing is an effective tool that provides insights into function behavior across an interval.
When sketching \( y = \operatorname{coth} x \) between \( x = 5 \) and \( x = 10 \), you should observe the smooth hyperbolic curve's approach to \( y = 1 \) as \( x \) increases.
This graphical approach helps explain why the integral approximate value from 5 to 10 is close to \( 5 \). Since \( \operatorname{coth} x \) is near \( 1 \) in this interval, multiplying the interval length, which is also \( 5 \), results in a product close to the area under the curve. Graphical analysis thus links visual interpretation with numerical estimation, revealing the interplay between algebraic expressions and their geometric embodiments.

Numerical Approximation

Numerical approximation is a powerful method in calculus for estimating the area under curves when exact solutions are complex or unnecessary. When evaluating \( \int_{5}^{10} \operatorname{coth} x \, dx \) graphically, the approximation method assumes that \( \operatorname{coth} x \) remains almost constant at \( 1 \) over the interval.
This results in an estimated area close to \( 5 \). However, more precise methods, like integration formulas or computational tools, can provide a more accurate result, such as \( 4.04352 \), which refines our approximation.
Numerical approximation methods, including Riemann sums or trapezoidal rule, break the problem into manageable computation steps, sacrificing some precision for simplicity and practical results. It is crucial to verify approximation accuracy through analytical methods, as demonstrated by calculating the error margin of approximately \( 0.95648 \) between the rough and refined integral values.

Antiderivatives

Antiderivatives are the counterpart to derivatives in calculus, representing the inverse operation to differentiation. To solve a definite integral like \( \int_{5}^{10} \operatorname{coth} x \, dx \) analytically, finding an antiderivative of \( \operatorname{coth} x \) is essential.
The derivative of \( \operatorname{coth} x \) is \( -\operatorname{csch}^2(x) \). By reversing this process, we identify the antiderivative as part of our integration process:

• \( \int \operatorname{coth} x \, dx = -\operatorname{csch}^2(x) + C \)

where \( C \) represents an arbitrary constant, not used in definite integrals.
By evaluating this expression at the bounds \( x = 5 \) and \( x = 10 \), you can compute the actual accumulated area under the \( \operatorname{coth} x \) curve. Calculating this difference provides an accurate measure of the integral, \( 4.04352 \), enabling verification against our numerical approximation and offering an understanding of the systematic derivation of an integral’s solution.