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Definite Integrals

Definite Integrals

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.

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