/*! 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 22 Find the length of the arc of th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 22

Find the length of the arc of the parabola \(y=x^{2}\) that lies between \((-1,1)\) and \((1,1)\).

Short Answer

Expert verified
The arc length is approximately 2.9578.

Step by step solution

01

Understand the Arc Length Formula

The formula for the arc length of a curve given by the equation \(y=f(x)\) from \(x=a\) to \(x=b\) is given by \[L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}}\, dx.\]In this problem, \(f(x) = x^2\), and we are asked to find the arc length from \(x = -1\) to \(x = 1\).
02

Differentiate the Function

To use the arc length formula, we first need to compute \(\frac{dy}{dx}\). Given \(y = x^2\), the derivative is \(\frac{dy}{dx} = 2x\).
03

Integrate to Find the Arc Length

Substitute \(\frac{dy}{dx} = 2x\) into the arc length formula:\[L = \int_{-1}^{1} \sqrt{1 + (2x)^2} \, dx = \int_{-1}^{1} \sqrt{1 + 4x^2} \, dx.\]This is the integral we need to evaluate to find the arc length.
04

Solve the Integral

The integral \(\int \sqrt{1 + 4x^2} \, dx\) is a standard integral that can be solved using the trigonometric substitution \(x = \frac{1}{2}\tan(\theta)\), leading to:\[L = \frac{1}{2}\int \sec^3(\theta) \, d\theta.\]After evaluating, converting back to terms of \(x\), and computing the definite integral from \(x = -1\) to \(x = 1\), approximately, the arc length is \(2.9578\).
05

Final Answer

After evaluating the integral, the calculated arc length between ((-1,1)) and ((1,1)) on the parabola \(y=x^2\) is approximately \(2.9578\).

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!

Key Concepts

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

Parabola
A parabola is a unique type of curve on a graph that is symmetric and U-shaped. The simplest form of a parabola is given by the equation \(y = x^2\). This curve opens upwards and its vertex, or lowest point, is at the origin (0,0).
  • Parabolas can open upwards or downwards depending on the coefficient of \(x^2\). If positive, like in \(y = x^2\), the parabola opens up. If negative, it opens down.
  • The axis of symmetry for a parabola defined by \(y = ax^2\) is the line \(x = 0\).
  • The range of a parabola opening upwards is \([0, \infty)\).
For the given exercise, we focus on the portion of the parabola between the points (-1,1) and (1,1). Understanding the shape and properties of a parabola helps visualize the segment of the arc we calculate.
Trigonometric Substitution
Trigonometric substitution is a technique used in calculus to simplify integrals that involve square roots. It involves substituting a trigonometric function for a variable to eliminate square roots.
  • This method is particularly helpful in solving integrals of the form \(\int \sqrt{a^2 + b^2x^2} \, dx\).
  • Common substitutions include \(x = a \sin(\theta)\), \(x = a \tan(\theta)\), or \(x = a \sec(\theta)\), depending on the integral.
In our solution, to evaluate \(\int \sqrt{1 + 4x^2} \, dx\), we use the substitution \(x = \frac{1}{2}\tan(\theta)\). This transforms the integral into one involving \(\sec^3(\theta)\), which is simpler to integrate.
Definite Integral
A definite integral calculates the exact area under a curve between two points along the x-axis. Unlike indefinite integrals, which yield a general formula, definite integrals provide a numeric value that represents this "area."
  • The notation \(\int_{a}^{b} f(x) \, dx\) indicates integration of function \(f(x)\) from \(x = a\) to \(x = b\).
  • For solving, compute an antiderivative of \(f(x)\) and evaluate it from \(x = a\) to \(x = b\).
In our arc length problem, the definite integral \(\int_{-1}^{1} \sqrt{1 + 4x^2} \, dx\) gives the total length of the curve between the points \((-1,1)\) and \((1,1)\) on the parabola \(y = x^2\).
Derivative Calculation
Derivatives measure the rate at which a function changes. They are fundamental in calculus because they provide a way to understand the shape of graphs and solve various real-world problems.
  • The derivative of a function \(f(x)\) is denoted as \(f'(x)\) or \(\frac{df}{dx}\).
  • For the function \(y = x^2\), the derivative is computed as \(\frac{dy}{dx} = 2x\).
  • This derivative tells us the slope of the tangent to the curve at any point \(x\).
Using this derivative in the arc length formula allows us to account for how the curve bends and changes direction from point to point, which is essential for calculating the length of a curved path.

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

Differentiate with respect to \(x\) : (a) \(\sinh 3 x\) (b) \(\tanh 4 x\) (c) \(x^{3} \cosh 2 x\) (d) \(\ln \left(\cosh \frac{1}{2} x\right)\) (e) \(\cos x \cosh x\) (f) \(1 / \cosh x\)

Differentiate the function \(f\) where \(f(x)\) is (a) \(x^{9}\) (b) \(\sqrt{\left(x^{3}\right)}\) (c) \(-4 x^{2}\) (d) \(4 x^{4}+2 x^{5}\) (e) \(4 x^{3}+x-8\) (f) \(1 /\left(2 x^{2}\right)\) (g) \(x+\sqrt{x}\) (h) \(2 x^{7 / 2}\) (i) \(1 /\left(3 x^{3}\right)\)

Express the length of the are of the curve \(y=\sin x\) from \(x=0\) to \(x=\pi\) as an integral. Also find the volume of the solid generated by revolving the region bounded by the \(x\) axis and this are about, the \(x\) axis through \(2 \pi\) radians.(b) Show that the curved surface generated by the revolution about the \(x\) axis of the portion of the curve \(y^{2}=4 a x\) included between the origin and the ordinate \(x=3 a\) is \(\frac{56}{3} \pi a^{2}\)

Find the stationary values of the following. functions, distinguishing carefully between them. In each case sketch a graph of the function. (a) \(f(x)=\frac{3 x}{(x-1)(x-4)}\) (b) \(f(x)=2 \mathrm{e}^{-\mathrm{s}}(x-1)^{3}\) (c) \(f(x)=x^{2} \mathrm{e}^{-3}\) (d) \(f(x)=\frac{1}{x^{2}}+\frac{8}{(1-x)^{2}}\)

A tank is initially filled with 1000 litres of brine, containing \(0.15 \mathrm{~kg}\) of salt per litre. Fresh brine containing \(0.25 \mathrm{~kg}\) of salt per litre runs into the tank at a rate of 4 litres \(\mathrm{s}^{-1}\), and the mixture (kept uniform by vigorous stirring) runs out at the same rate. Show that if \(Q\) (in \(\mathrm{kg}\) ) is the amount of salt in, the tank at time \(t\) (in \(s\) ) then $$ \frac{\mathrm{d} Q}{\mathrm{~d} t}=1-\frac{Q}{250} $$
See all solutions

Recommended explanations on Physics Textbooks

Kinematics Physics

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Electromagnetism

Read Explanation

Particle Model of Matter

Read Explanation

Torque and Rotational Motion

Read Explanation

Electricity and Magnetism

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.