Suppose that f is a continuous function defined on [a, b].

Case 1. f(x) > 0.
From the examples f(x) = sin(x) and f(x) = x2, it appears reasonable to define the area bounded by the graph of y = f(x), the x-axis, the lines x = a and x = b to be the definite integral

Case 2. f(x) < 0.
In forming the Riemann sum, the value of f(xi) < 0 and, hence, f(xi) Dx < 0. It follows that each Riemann sum

Using properties of limits, we obtain

To get the area of the ith approximating rectangle, we need to multiply the quantity f(xi) Dx by - 1. Consequently, it appears reasonable to define the area bounded by the graph of y = f(x) , the x-axis, and the lines x = a and x = b to be - 1 times the definite integral of f:

Case 3. Arbitrary continuous functions f
If the range of f contains both positive and negative values, the integral of f is no longer the area. For example,

which is not the area bounded by the graph of y = sin(x) and the x-axis between - p and p. The integral is the sum of - (the area from - p to 0) and the area from 0 to p; these terms cancel each other out since the sine function is an odd function.

To find the area bounded by f, we use the absolute value of f. For example,

is the area bounded by the graph of the sine function and the x-axis from - p to p.