Chapter 5.1, Problem 64E

### Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

Chapter
Section

### Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
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,Â y=xx2+1

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

mx=xx2+1

Multiply by x2+1

mxÂ·x2+1=xx2+1Â·x2+1

By simplifying,

mx3+mx=x

Subtract by x.

mx3+mx-x=0

Simplify.

xmx2+m-1

Simplify.

x=0Â orÂ mx2+m-1

Solve for x by quadratic formula.

x=Â±1m-1

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

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

Area =âˆ«-kkxx2+1-mxdx

By using âˆ«-aaf(x)dx=2âˆ«0af(x)dx

A=2âˆ«0kxx2+1

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