For example: find the area of the region bounded by the x-axis, y= 0, and the parabola, \(\displaystyle y= x^2- 1\).

The first thing you should do is draw the graph- or at least imagine it- to see that the two cross at x= -1 and x= 1, that the only region bounded by those is between x= -1 and x= 1, and that, in that region, the x-axis is always **above** the parabola.

To find the area, subtract the equation of the **lower** curve from the equation of the higher curve, 0- (x^2- 1)= 1- x^2, and integrate from x= -1 to x= 1:

\(\displaystyle \int_{-1}^1 1- x^2 dx= \)\(\displaystyle \left[ x- \frac{x^3}{3}\right]_{-1}^1=\)\(\displaystyle (1- \frac{1}{3})- (-1+ \frac{1}{3})\)\(\displaystyle = 2- \frac{2}{3}= \frac{4}{3}\).

Of course, "area" of any region is a **positive** number and we guaranteed that by subtracting "the equation of the **lower** curve from the equation of the higher curve".