Barrons AP Calculus - David Bock

FIGURE N6–3

Note 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 dx 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.
E1. Using Rectangles.
We may approximate by any of the following sums, where Δx represents the

subinterval widths:
(1) Left sum: f (x 0 ) Δx 1 + f (x 1 ) Δx 2 + ... + f (xn − 1) Δxn, using the value of f at the left endpoint of
each subinterval.
(2) Right sum: f (x 1 ) Δx 1 + f (x 2 ) Δx 2 + ... + f (xn) Δxn, using the value of f at the right end of each
subinterval.
(3) Midpoint sum: using the value of f at the midpoint
of each subinterval.
These approximations are illustrated in Figures N6–4 and N6–5, which accompany Example 24.
EXAMPLE 24
Approximate by using four subintervals of equal width and calculating:
(a) the left sum,