Question

Find the number b such that the line y = b divides the region bounded by the curves y = 25x2 and y = 4 into two regions with equal area. (round your answer to two decimal places.

Answer 1

$y=25x^2\implies x=\pm5\sqrt y$

The area between the parabola and the line is

$\displaystyle\int_0^4(5\sqrt y-(-5\sqrt y))\,\mathrm dy=10\int_0^4\sqrt y\,\mathrm dy=\dfrac{10}{\frac32}y^{3/2}\bigg|_{y=0}^{y=4}=\dfrac{20}3\times8=\frac{160}3$

But you're looking for the line $y=b$ such that the areas of each subdivision are equal. This means you want $b$ to satisfy

$\displaystyle10\int_0^b\sqrt y\,\mathrm dy=\dfrac{80}3$

You have

$\displaystyle\int_0^b\sqrt y\,\mathrm dy=\dfrac83$
$\dfrac23y^{3/2}\bigg|_{y=0}^{y=b}=\dfrac83$
$\dfrac23b^{3/2}=\dfrac83$
$b^{3/2}=4$
$b=4^{2/3}=2\sqrt[3]2\approx2.52$