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Chapter 8: Problem 13

\(7-18\) Find the length of the curve. $$y=\ln (\sec x), \quad 0 \leqslant x \leqslant \pi / 4$$

Short Answer

Expert verified
The length of the curve is \( \ln(\sqrt{2} + 1) \).

Step by step solution

01

Understand the Arc Length Formula

The formula for the arc length of a curve described by the function \( y = f(x) \) over the interval \([a, b]\) is given by \( L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \). In this problem, we have \( y = \ln(\sec x) \) and need to find the arc length from \( x = 0 \) to \( x = \frac{\pi}{4} \).
02

Compute the Derivative \( \frac{dy}{dx} \)

First, find the derivative of \( y = \ln(\sec x) \). We will use the chain rule: \( \frac{d}{dx}[\ln(\sec x)] = \frac{1}{\sec x} \cdot \frac{d}{dx}[\sec x] \). The derivative of \( \sec x \) is \( \sec x \tan x \), so \( \frac{dy}{dx} = \tan x \).
03

Plug into Arc Length Formula

Now substitute \( \frac{dy}{dx} = \tan x \) into the arc length formula: \[ L = \int_0^{\pi/4} \sqrt{1 + (\tan x)^2} \, dx \]. Simplifying under the square root, recall the Pythagorean identity \( 1 + \tan^2 x = \sec^2 x \). Hence, \( L = \int_0^{\pi/4} \sec x \, dx \).
04

Integrate to Find the Arc Length

The integral \( \int \sec x \, dx \) can be solved by recognizing that \( \int \sec x \, dx = \ln |\sec x + \tan x| + C \). Evaluating from \( x = 0 \) to \( x = \frac{\pi}{4} \), we get \[ L = \left[ \ln |\sec x + \tan x| \right]_0^{\pi/4} = \ln(\sec(\pi/4) + \tan(\pi/4)) - \ln(\sec(0) + \tan(0)) \].
05

Simplify and Calculate the Final Answer

Compute the necessary values: \( \sec(\pi/4) = \sqrt{2} \) and \( \tan(\pi/4) = 1 \), so \( \sec(\pi/4) + \tan(\pi/4) = \sqrt{2} + 1 \). Also, \( \sec(0) = 1 \) and \( \tan(0) = 0 \), so \( \sec(0) + \tan(0) = 1 \). Therefore, \[ L = \ln(\sqrt{2} + 1) - \ln(1) = \ln(\sqrt{2} + 1) \].
06

Final Answer

The length of the curve \( y = \ln(\sec x) \) from \( x = 0 \) to \( x = \frac{\pi}{4} \) is \( \ln(\sqrt{2} + 1) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integral Calculus
Integral calculus is a fundamental branch of calculus that revolves around the concept of integration. It is primarily concerned with calculating the area under curves, among other applications. In this exercise, integral calculus is applied to determine the arc length of a curve.

The arc length formula for a curve defined by a function \( y = f(x) \) over an interval \([a, b]\) is given by:
  • \[ L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
The goal is to evaluate this integral to find a numerical description of curvature distance.

In this problem, integration must be performed over an interval from \( x = 0 \) to \( x = \frac{\pi}{4} \), as dictated by the exercise.
Chain Rule
The chain rule is a pivotal technique in derivative calculus used to find the derivative of composite functions. It allows us to differentiate functions nested within one another efficiently.

In this example, we need the derivative of \( y = \ln(\sec x) \). Here, \( \sec x \) is the inside function contained within the logarithmic function. Applying the chain rule involves two main steps:
  • First, differentiate the outer function, the logarithm: \( \frac{d}{dx}[\ln(u)] = \frac{1}{u} \).
  • Next, differentiate the inner function, \( \sec x \), to obtain \( \frac{d}{dx}[\sec x] = \sec x \tan x \).
This results in:
  • \( \frac{dy}{dx} = \frac{1}{\sec x} \cdot \sec x \tan x = \tan x \).
The chain rule thus facilitates the calculation of the derivative necessary for our arc length formula.
Derivative of Logarithmic Functions
Understanding the derivative of logarithmic functions is crucial as they frequently appear in calculus problems. The derivative of a natural logarithm function, \( \ln(u) \), is derived using the transformation of the original function.

The derivative of \( \ln(u) \) of a function \( u(x) \) is:
  • \( \frac{d}{dx}[\ln(u)] = \frac{1}{u} \cdot \frac{du}{dx} \)
This formulation demands that you differentiate \( u(x) \) separately, as seen in this problem where \( u(x) = \sec x \).

This principle turns into real application by plugging in the derived derivative back into the arc length formula. Calculating \( \frac{dy}{dx} = \tan x \) sets up the integral for solving arc length issues with precision.
Trigonometric Identities
Trigonometric identities are indispensable tools in mathematics, providing relationships between trigonometric functions. In this exercise, the Pythagorean identity plays an essential role.

The identity \( 1 + \tan^2 x = \sec^2 x \) is used to simplify computations under the square root within the integral calculus equation for arc length. This simplification process turns the integral:

\[ \sqrt{1 + (\tan x)^2} = \sqrt{\sec^2 x} = \sec x \]

This transformation simplifies the integrand, making the expression much more manageable.
  • \( L = \int_0^{\pi/4} \sec x \, dx \)
Using trigonometric identities thus reduces complex expressions, aiding in finding solutions straightforwardly.

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Most popular questions from this chapter

Find the arc length function for the curve \(y=2 x^{3 / 2}\) with starting point \(P_{0}(1,2) .\)

The speeds of vehicles on a highway with speed limit 100 \(\mathrm{km} / \mathrm{h}\) are normally distributed with mean 112 \(\mathrm{km} / \mathrm{h}\) and standard deviation 8 \(\mathrm{km} / \mathrm{h}\) . (a) What is the probability that a randomly chosen vehicle is traveling at a legal speed? (b) If police are instructed to ticket motorists driving 125 \(\mathrm{km} / \mathrm{h}\) or more, what percentage of motorists are targeted?

\(23-26\) Use Simpson's Rule with \(n=10\) to estimate the arc length of the curve. Compare your answer with the value of the integral produced by your calculator. $$y=x \ln x, \quad 1 \leqslant x \leqslant 3$$

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The hydrogen atom is composed of one proton in the nucleus and one electron, which moves about the nucleus. In the quan- tum theory of atomic structure, it is assumed that the electron does not move in a well-defined orbit. Instead, it occupies a state known as an orbital, which may be thought of as a "cloud of negative charge surrounding the nucleus. At the state of lowest energy, called the ground state, or Is-orbital, the shape of this cloud is assumed to be a sphere centered at the nucleus. This sphere is described in terms of the probability density function $$p(r)=\frac{4}{a_{0}^{3}} r^{2} e^{-2 r / a_{0}} \quad r \geqslant 0$$ where \(a_{0}\) is the Bohr radius\(\left(a_{0} \approx 5.59 \times 10^{-11} \mathrm{m}\right) .\) The integral $$P(r)=\int_{0}^{r} \frac{4}{a_{0}^{3}} s^{2} e^{-2 s / a_{0}} d s$$ gives the probability that the electron will be found within the sphere of radius \(r\) meters centered at the nucleus. (a) Verify that \(p(r)\) is a probability density function. (b) Find lim_{r\rightarrow\infty} \(p(r) .\) For what value of \(r\) does \(p(r)\) have its maximum value? (c) Graph the density function. (d) Find the probability that the electron will be within the sphere of radius 4\(a_{0}\) centered at the nucleus. (e) Calculate the mean distance of the electron from the nucleus in the ground state of the hydrogen atom.
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