Question

Find the area of a function using integrals that equals .5 (can range from .499 to .504)
There has to be three regions for the equations

Answer 1

Are you looking for a piecewise function? If so, a simple step function will do the job.

f(x) =
0.1 for 0 <= x < 1
0.2 for 1 <= x < 2
0.4 for 2 <= x <= 2.5

You can do similar things with sloping lines, but you will get trapezoids or triangles to work with.

If you want a single function, that's a bit tougher, since you want three regions. A cubic may have some area below the x-axis, so you could divide the whole thing into three regions, integrating each one separately to add the negative area, rather than subtracting it.

So, just picking a simple cubic and integrating its absolute value, I got a result of 0.89

Scaling it by .5/.89, the final area is very close to 0.5:

http://www.wolframalpha.com/input/?i=.5%2F.89+%E2%88%AB%5B-1,3%2F2%5D+%7C(x-1)(x%2B1)(x)%7C+dx

To make that three regions, get rid of the || signs, and split the integral into three parts, divided at the roots:

0.5/0.89 times the sum of
?[-1,0] (x-1)(x+1)(x) dx
+ ?[0,1] -(x-1)(x+1)(x) dx
+ ?[1,3/2] (x-1)(x+1)(x) dx
= 0.5/0.89 (1/4 + 1/4 + 0.39)
= 0.5/0.89 (0.89)
= 0.5