   Chapter 5.P, Problem 2P

Chapter
Section
Textbook Problem

There is a line through the origin that divides the region bounded by the parabola y = x − x 2 and the x -axis into two regions with equal area. What is the slope of that line?

To determine

To find:

The slope of the line satisfying the given condition.

Explanation

1) Given:

Equation of parabolais y=x-x2.

2) Explanation:

The equation of the parabola is y=x-x2

Solve x-x2=0,

x1-x=0

Therefore, x=0 or x=1.

The area of the region bounded by the parabola and x-axis is

01x-x2dx=x22-x3310=16

Suppose the slope of unknown line is m.

Then the equation of line is y=mx.

To find the point of intersection of parabola and line, solve x-x2=mx.

x2+mx-x=0

x2+ xm-1=0

x x+m-1=0

Therefore, either x=0 or x+m-1=0

So,

x=0 or 1-m

The area of the region above the line mx but below the parabola is half of total area of the region bounded by the parabola and  x-axis

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