/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 53 Find the arc length of the curve... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 53

Find the arc length of the curve over the given interval. $$ y=\ln x, \quad[1,5] $$

Short Answer

Expert verified
The arc length of the given function over the interval \([1,5]\) is \(\sinh^{-1}(5) - \sinh^{-1}(1).\)

Step by step solution

01

Find the Derivative of the Function

The given function is \(y = \ln x\). Its derivative, obtained using the rule \(d/dx \ln x = 1/x\), is therefore, \(y' = 1/x\).
02

Substitute the Derivative in the Formula and Evaluate the Integral

Now, we need to substitute \(y'\) into the arc length formula. So, we get \(L = \int_1^5\sqrt{1+(1/x)^2}dx.\) This integral may not be straightforward to solve, a good way would be to simplify the integrand as follows: \[L =\int_1^5 \frac{\sqrt{x^2+1}}{x} dx.\] To solve this punctual integral without exhaustive steps, recognize the integrand as the derivative of arcsinh(x) - i.e, the hyperbolic arcsine of x. Then, L becomes a difference of arcsinh(x) over the interval.
03

Compute the Arc Length Using the Integral Limits

In this final step, let's use the fundamental theorem of calculus which states that, if a function F(x), is the integral of f(x), over the domain [a,b], then 鈭� from a to b of f(x) dx equals F(b) - F(a). Because the integral from step 2 is [arcsinh(x)] from 1 to 5, we can simply substitute these limits to get, \[L = \sinh^{-1}(5) - \sinh^{-1}(1)\].

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Sketch the graph of \(g(x)=\left\\{\begin{array}{ll}e^{-1 / x^{2}}, & x \neq 0 \\ 0, & x=0\end{array}\right.\) and determine \(g^{\prime}(0)\).

Use mathematical induction to verify that the following integral converges for any positive integer \(n\). \(\int_{0}^{\infty} x^{n} e^{-x} d x\)

The region bounded by \((x-2)^{2}+y^{2}=1\) is revolved about the \(y\) -axis to form a torus. Find the surface area of the torus.

Use a computer algebra system to find the integral. Graph the antiderivatives for two different values of the constant of integration. $$ \int \sec ^{5} \pi x \tan \pi x d x $$

(a) find the indefinite integral in two different ways. (b) Use a graphing utility to graph the antiderivative (without the constant of integration) obtained by each method to show that the results differ only by a constant. (c) Verify analytically that the results differ only by a constant. $$ \int \sec ^{2} x \tan x d x $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.