Figure N6–3Note that if f is continuous then the area between the graph of f on [a,b] and
the x-axis is given by
.This implies that, over any interval within [a,b] for which f(x) < 0 (for which its
graph dips below the x-axis), |f(x)| = −f(x). The area between the graph of f and
the x-axis in Figure N6–3 equals
.This topic is discussed further in Chapter 7.E. APPROXIMATIONS OF THE DEFINITE INTEGRAL;
RIEMANN SUMS
It is always possible to approximate the value of a definite integral, even when
an integrand cannot be expressed in terms of elementary functions. If f is
nonnegative on [a,b], we interpret as the area bounded above by y = f(x),
below by the x-axis, and vertically by the lines x = a and x = b. The value of the
definite integral is then approximated by dividing the area into n strips,
approximating the area of each strip by a rectangle or other geometric figure,
then summing these approximations. We often divide the interval from a to b
into n strips of equal width, but any strips will work.