Jason K.

asked • 03/04/16

Find the number

Find the number b such that the line y = b divides the region bounded by the curves y = x2 and y = 4 into two regions with equal area. Give your answer correct to 3 decimal places

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Kenneth S. answered • 03/04/16

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b = 4^(2/3) = 2.520
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Kenneth S.

Without loss of generality, because of symmetry of the parabola y = x squared, we can draw & shade a region in the first quadrant whose boundary line at top is y = 4, and whose lower boundary is y = x squared, which runs with upward concavity from the origin up to (2,4). The third boundary is the y axis.  This enclosed area is called Region Z.
 
Now draw a horizontal line y = b crossing the y-axis around 2.5 (just an estimate, in order to make a reasonable sketch.
Call the area within Region Z and below the y = b line A1, and the rest of Region Z above it is designated Area A2.
 
A1 = def integral from x = 0 to x = sqrt (b); integrand is (b - x2) = (2/3)b^(3/2).
 
A2 = R - A1 and R = def integral from x to 4 of integrand (4 - x2) = 16/3.
 
Setting A1 = A2     or A1 = R - A1, we come to the solution b = 4^(2/3).
 
 
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03/04/16

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