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Q37E

Expert-verified
Found in: Page 370

### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

# 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.

The value of $$b$$ is $$2.52$$.

See the step by step solution

## The area under a curve

The area under a curve between two points is found out by doing a definite integral between the two points.

## Find $$b$$ such that the blue and red areas are equal.

The regions $$y = {x^2},y = 4$$ are divided by $$y = b$$

Integrate along the $$y$$-axis, so for $$y = {x^2}$$ use $$x = \sqrt y$$, which is the right half of the parabola. The right/left halves are symmetric so ignore the left half.

The red region is the integral from $$y = 0$$ to $$b$$, the blue region is from $$y = b$$ to 4

## Set the two integral areas equal to each other to solve for $$b$$

The value of $$b$$ is given by

$$\int_0^b {\sqrt y \,} dy = \int_b^4 {\sqrt y \,} dy$$

$\left[{\frac{2}{3}{y^{\frac{3}{2}}}} \right]_0^b = \left[ {\frac{2}{3}{y^{\frac{3}{2}}}} \right]_b^4$

$${b^{\frac{3}{2}}} - 0 = {4^{\frac{3}{2}}} - {b^{\frac{3}{2}}}$$

$$2{b^{\frac{3}{2}}} = 8$$

$${b^{\frac{3}{2}}} = 4$$

$$b = {4^{\frac{2}{3}}}$$

$$\approx 2.52$$