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:
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.