Find the number b such that the line y = b divides the region bounded by the curves y = 25x2 and y = 1 into two regions with equal area. (Round your answer to two decimal places.)

y=1/∛4 divides the area in half

Step-by-step explanation:

since the minimum value of x² is 0 (for x=0) and for y=1

1 = 25*x² → x = ±√ (1/25) = ±1/5

then the total area between y=1 and y = 25*x² is bounded to x=±1/5 and y=0. Since there is a direct relationship between x and y, we can find the value of x=a that divides the region in 2 of the same area. thus

Area below x=C = Area above x=C

Area below x=C = Total area - Area below x=C

2*Area below x=C = Total area

Area below x=C = Total area / 2

∫ 25*x² dx from x=c to x=-c = 1/2 ∫ 25*x² dx from x=1/5 to x=-1/5

25*[c³/3 - (-c) ³/3] = 25/2 * [ (1/5) ³/3 - (-1/5) ³/3]

2*c³/3 = (1/5) ³/3

c = 1 / (5*∛2)

thus

y=25 * x² = 25*[1 / (5*∛2) ]² = 1/∛4

thus the line y=1/∛4 divides the area in half