What if f(x) is negative? Then any area above the graph and below the x-axis
is counted as negative (Figure N6–2).
Geometrically, area is always positive, so the shaded area above the curve and
below the x-axis equals
,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.
Figure N6–2We 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) on the previous pages,
,therefore
.