(Round your answer to two decimal places.)
Find the number b such that the line y = b divides the region bounded by the curves y = 4x^2 and y = 9 into two regions with equal area.
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Since either region is bounded by the parabola y=4x2 and a horizontal line (y=b and y=9), you should integrate the function along the y-axis, not along the x-axis as usual. Solve the function for x and get x=½√y for x≥0. Due to the symmetry of the parabola, we do not need to integrate the other half of the parabola, x=-½√y for x≤0. We know that the area inside the parabola between y=0 and y=b is supposed to be half of the area inside the parabola between y=0 and y=9, which means
∫0b (½√y)dy = ½ ∫09 (½√y)dy.
Evaluate these two integrals and solve for b. You will get b=9/22/3≈5.67.
What we need is two integrals, one evaluated from 0 to b, and the other from b to 9, set these equal to one another and try to find a value for b.
∫b0 4x2dx = ∫9b 4x2dx = 4x3/3
If we evaluate the integral between 0 and 9 we get a total area of 972, so we need to choose b so that
∫b0 4x2dx = ∫9b 4x2dx = 486
(4b3/3) = 486
b3 = (729/2) = 364.5
b = (364.5)(1/3) = 7.14
Y = 4 X ^ 2
Y = 9
Y axis is the axis of Symetry
4X^2 - 9 = 0
X = ±3/2
Area right of the curve:
∫ 4X^2 dx = 4X^3 l = 4/3 . 27/ 8 = 9/2.
0 3 l 0
Area to the left = 9 . 3/2 - 9/2 = 18/2 = 9
The same area is to the left of the Y-axis( Axis of Symmetry.