Negative function : Several functions are not positive however.  Consider the case of f (x ) =x² - 4 on [0,2].  If we utilizes n = 8 and the midpoints for the rectangle height we obtain the following graph,

In this case let's notice that the function lies entirely below the x-axis and therefore is always negative.  If we avoid the fact that the function is negative always and utilizes the same ideas above to estimate the area among the graph and the x-axis we get,

A_m=  (1/4) f(1/8) +(1/4) f(3/8)+(1/4) f(5/8)+(1/4) f(7/4)+(1/4)f (9/8)+(1/4) f(11/8) +(1/4) f(11/8)+(1/4) f(13/8)+(1/4) f(15/8)  

= -5.34375

Our answer is negative as we have expected given that all the function evaluations are negative.