# Definite Integrals of Real-Valued Functions

When we're integrating a non-negative function from *a* to *b*, the integral **can be thought of as the "area under the curve" of the function. However, most of the time we can't count on having a non-negative function to integrate.**

Assume *f* is a function that's allowed to take on negative values, and we're integrating from *a* to *b* with *a *< *b*. Then is the weighted sum of the areas between the graph of *f* and the *x*-axis. We look at all areas between *f* and the *x*-axis. If they're on top of the *x*-axis we count them positively. If they're below the *x*-axis we count them negatively.

In other words, we add all the areas on top of the *x*-axis, then subtract all the areas below the *x*-axis.