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Easy: A parabola and the x-axis
Set the two functions equal to find where they cross; those x-values are the limits of integration. On the interval, decide which function is on top by testing a point between them. The area is the integral of (top − bottom), which is always positive.
Find the area of the region enclosed by the two curves.
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Test a point between the intersections. Whichever function gives the larger value there is on top for that whole interval.
You subtracted top from bottom the wrong way. Area between curves is ∫(top − bottom) dx, which is always positive.
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