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katrin [286]
3 years ago
14

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 equ

al area. (round your answer to two decimal places.
Mathematics
1 answer:
umka2103 [35]3 years ago
4 0
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
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