- Case 1.
**f(x)**.__>__0 - From the examples
**f(x) = sin(x)**and**f(x) = x**, it appears reasonable to define the area bounded by the graph of^{2}**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(x**and, hence,_{i})__<__0**f(x**. It follows that each Riemann sum_{i}) Dx__<__0Using properties of limits, we obtain

To get the area of the i

^{th}approximating rectangle, we need to multiply the quantity**f(x**by_{i}) Dx**- 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.