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 N7–2
See Question 11 in the Practice Exercises.
If x is given as a function of y, say x = g(y), then (Figure N7–3) the subdivisions are made along
the y-axis, and the area bounded by the y-axis, the curve, and the horizontal lines y = a and y = b is
given exactly by
See Questions 3 and 13 in the Practice Exercises.FIGURE N7–3
A1. Area Between Curves.
To find the area between curves (Figure N7–4), we first find where they intersect and then write the
area of a typical element for each region between the points of intersection. For the total area
bounded by the curves y = f (x) and y = g(x) between x = a and x = e, we see that, if they intersect at
[c,d], the total area is given exactly by