   Chapter 5.1, Problem 57E

Chapter
Section
Textbook Problem

# Find the number b such that the line y   = b divides the region bounded by the curves y   =   x 2 and y   =   4 into two regions with equal area.

To determine

To find:

The number b such that the line y=b divides the region bounded by the curves y=x2 and y=4 into two regions with equal area

Explanation

1) Concept:

Find b such that the blue and light green areas are equal.

2) Given:

The curves y=x2, y=4 divided into two regions with equal area

3) Formula:

xn dx=xn+1n+1+C

4) Calculation:

Given that y=x2, y=4 and region is divided by y=b.

Find b such that the blue and light green areas are equal.

Integrate along y- axis.

For the parabola y=x2,

Take square root of both sides.

x=y

This is the right half of the parabola.

The right and left halves are symmetric.

The blue region is the integral from y=0 to b, and the light green region is from y=b to 4.

Set the blue and light green areas as equal, and then solve for b.

0bydy=b4ydy

Solve integrals.

23y320b=23y32b4

Substitute limits.

23b32-23032=23432-23b32

Divide by 23, and then simplify

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