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In Section we motivated the definition of the definite integral with the idea of finding
the area of a region in the plane. However, to solve the problem we restricted to a very
special type of region, namely, a region lying between the graph of a function f and an
interval on the x-axis. We will now consider the more general problem of the area of a
region lying between the graphs of two functions.
y = f( x )
y = g( x )
a b
Figure Approximating the area between y = f(x) and y = g(x)
Suppose f and g are functions defined on an interval [a, b] with g(x) ≤ f(x) for all x
in [a, b]. We suppose that f and g are integrable on [a, b], from which it follows that the
function k defined by
k(x) = f(x) − g(x)
is also integrable on [a, b]. Let R be the region lying between the graphs of f and g over
the interval [a, b] and let A be the area of R. In other words, A is the area of the region of
the plane bounded by the curves y = f(x), y = g(x), x = a, and x = b. We begin with an
approximation for A. First, we divide [a, b] into n intervals of equal length
b − a
∆x =
n
and let a = x0 < x1 < x2 < x3 < · · · < xn = b be the endpoints of these intervals. Next,
for i = 1, 2, 3, . . . , n, let Ri be the region lying between the graphs of f and g