(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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# 3 Answers

Since either region is bounded by the parabola y=4x

^{2}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∫

_{0}^{b}(½√y)dy = ½ ∫_{0}^{9}(½√y)dy.Evaluate these two integrals and solve for b. You will get b=9/2

^{2/3}≈5.67.Dear Ben,

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.

∫

^{b}_{0}4x^{2}dx = ∫^{9}_{b}4x^{2}dx = 4x^{3}/3If we evaluate the integral between 0 and 9 we get a total area of 972, so we need to choose b so that

∫

^{b}_{0}4x^{2}dx = ∫^{9}_{b}4x^{2}dx = 486(4b

^{3}/3) = 486b

^{3}= (729/2) = 364.5b = (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:

3/2 3/2

∫ 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.