   Chapter 7.8, Problem 43E

Chapter
Section
Textbook Problem

# Sketch the region and find its area (if the area is finite). S = { ( x , y ) | x ≥ 1 ,   0 ≤ y ≤ 1 / ( x 3 + x ) }

To determine

To sketch:

the given region, and find its area if the area is finite.

Explanation

Given:

S={(x,y)|x1,0y1/(x3+x)}

Formulae used: Area under the curve y=f(x) in the interval [a,b] is A=abf(x)dx

Consider S={(x,y)|x1,0y1/(x3+x)}

Plot curves where x[1,],y[0,1/(x3+x)] and get,

Now, from the graph f(x)=1x3+x in the interval [1,] the area appear finite.

Use the formulae to get the area under the cure, as

A=11x3+xdx

Decompose the above expression into partials fractions and get,

1x3+x=Ax+Bx+Cx2+1=A(x2+1)+(Bx+C)(x)x3+x1=Ax2+A+Bx2+Cx0x2+0x+1=(A+B)x2+Cx+A

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