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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

Suppose the graph of a cubic polynomial intersects the parabola y = x 2 when x = 0 , x = a , and x = b , where 0 < a < b . If the two regions between the curves have the same area, how is b related to a ?

To determine

To find:

The relation between a and b

Explanation

1) Concept:

Using y=x2 and a general cubic polynomial, form the diagram, and then set up an integral for area, and find the relation.

2) Calculation:

The general form of cubic polynomial can be written as px3+qx2+rx+s

where p, q, r, and s    are reals.

The cubic polynomial passes through the origin. Therefore, the equation becomes

y=px3+qx2+rx

At intersection of parabola an polynomial,

px3+qx2+rx=x2

p x3+q-1x2+rx=0 Let the curve fx = px3+qx2+rx-x2

That is, fx=px3+q-1x2+rx

Since a  and  b are points of intersection for both curves,

f(0)=fa=fb=0  That is a,b and 0 are the roots of cubic polynomial fx.

So, the above function can be written as

fx=x  x-ax-b

This is the same as

fx=xx2-a+bx+ab=[x3-a+bx2+abx]

The two areas are equal.

For the second region, there is a change in position of the upper and lower curves

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