FIGURE N6–1
*It is not necessary that the subintervals be of equal length, but the formulation is generally simpler if they are.
What if f (x) is negative? Then any area above the graph and below the x-axis is counted as
negative (Figure N6–2).
The shaded area above the curve and below the x-axis equals
FIGURE N6–2
where the integral yields a negative number. Note that every product f (xk) Δx in the shaded region is
negative, since f (xk) is negative for all x between a and b.
We see from Figure N6–3 that the graph of f crosses the x-axis at c, that area A 1 lies above the x-
axis, and that area A 2 lies below the x-axis. Since, by property (5),
therefore