   Chapter 5.1, Problem 64E

Chapter
Section
Textbook Problem

For what values of m do the line y = m x and the curve y = x / ( x 2 + 1 ) enclose a region? Find the area of the region.

To determine

To find:

The values of m  so that the line y=mx and the curve y=xx2+1 enclose a region and find the area of the region

Explanation

1) Concept:

Area is the integral of the difference of two functions.

2) Calculation:

Given that

y=mx, y=xx2+1

To find the intersection points of the given curves, equate both to each other.

mx=xx2+1

Multiply by x2+1

mx·x2+1=xx2+1·x2+1

By simplifying,

mx3+mx=x

Subtract by x.

mx3+mx-x=0

Simplify.

xmx2+m-1

Simplify.

x=0 or mx2+m-1

Solve for x by quadratic formula.

x=±1m-1

For 0<m<1, the intersection of two functions will be at x=0, and at x=±k where k=1m-1.

Since the region is symmetric, there will be two equal areas.

Area =-kkxx2+1-mxdx

By using -aaf(x)dx=20af(x)dx

A=20kxx2+1

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Solve the equations in Exercises 126. 2x3x12x+3x+1=0

Finite Mathematics and Applied Calculus (MindTap Course List)

In Exercises 7-10, solve the equation. 3x210x+8=0

Calculus: An Applied Approach (MindTap Course List)

In Exercises 39-54, simplify the expression. (Assume that x, y, r, s, and t are positive.) 42. 5x6y32x2y7

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Fill in each blank: 72ft=yd

Elementary Technical Mathematics

3. Find the 80th term of the arithmetic sequence with first term -2 and common difference 3.

Mathematical Applications for the Management, Life, and Social Sciences

Evaluate the integral. 51. 1x4x2+1dx

Single Variable Calculus: Early Transcendentals

Convert 3,400 feet to miles.

Trigonometry (MindTap Course List) 