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In this section we learn the second part of the fundamental theorem and we use it to compute the derivative of an area function.

### The Fundamental Theorem of Calculus, Part II

#### Area Functions

An area function gives us the net area between the curve $$ and the $$-axis over the interval $$. Net area means the area under the curve (where the function, $$, is positive) minus the area above the curve (where the function, $$, is negative).
Use the graph of $$ to find the values of the area function

Use the graph of $$ to find the values of the area function
From $$ to $$ the curve is a semi-circle

#### FTC, Part II

The second part of the FTC tells us the derivative of an area function $$.

This conclusion establishes the theory of the existence of anti-derivatives, i.e., thanks to the FTC, part II, we know that every continuous function $$ has an anti-derivative.

Here are some detailed, lecture style videos on area functions:
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