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.