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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.

y = x ( x 2 + 1 ) ,   y = x 5 x ,   x 0

To determine

To find:

Approximate x-coordinates of the points of intersection of the given curves, and find the area of the region bounded by the curves.

Explanation

1) Concept:

The area A of the region bounded by the curves y=f(x), y=g(x) and the lines x=a and x=b is

A= abfx-gxdx

fx-gx=fx-gx when fxg(x)gx-fx when gxf(x)

2) Given:

y=xx2+12,   y=x5-x,  x0

3) Calculation:

fx=xx2+12 and gx=x5-x

i) To find the intersection point of the curves, draw the graph.

From the sketch, it is clear that the curves intersect at x=0 and x=1.052

ii) the upper curve is y=xx2+12 and the lower curve is y=x5-x

Therefore,

A=01.052xx2+12-(x5-x)dx

=01.052xx2+12-x5+xdx

Applying integration separately,

=01.052xx2+12dx-01.052x5dx+01

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