Figure N7–1If f (x) is nonnegative on [a,b], as in Figure N7–1, then f(xk) Δx can be
regarded as the area of a typical approximating rectangle, and the area bounded
by the x-axis, the curve, and the vertical lines x = a and x = b is given exactly by
.
See Questions 1, 5, and 10 in the Practice Exercises at the end of this chapter.
If f (x) changes sign on the interval (Figure N7–2), we find the values of x for
which f (x) = 0 and note where the function is positive, where it is negative. The
total area bounded by the x-axis, the curve, x = a, and x = b is here given exactly
by ,
where we have taken into account that f (xk) Δx is a negative number if c < x < d.
Figure N7–2See Question 11 in the Practice Exercises.