The Fundamental Theorem of Calculus has a shortcut version that makes finding the area under a curve a snap. Here it is. Let F be any antiderivative of the function f; then
![image0.png](https://www.dummies.com/wp-content/uploads/219768.image0.png)
With this version of the Fundamental Theorem, you can easily compute a definite integral like
![image1.png](https://www.dummies.com/wp-content/uploads/219769.image1.png)
You could get this area with two different methods that involve area functions. First, you could determine the area function for this parabola that begins sweeping out area at x = 2, and then compute that area function’s output when x = 3. Second, you could determine the area function for the parabola that begins sweeping out area at x = 0, and then use that area function to subtract the area from 0 to 2 from the area from 0 to 3.
The beauty of the shortcut theorem is that you don’t have to use an area function like
![image2.png](https://www.dummies.com/wp-content/uploads/219770.image2.png)
or any other area function.
You just find any antiderivative, F(x), of your function, and do the subtraction, F(b) – F(a). The simplest antiderivative to use is the one where C = 0. So, here’s how you use the theorem to find the area under the parabola from 2 to 3.
![image3.png](https://www.dummies.com/wp-content/uploads/219771.image3.png)
and thus,
![image4.png](https://www.dummies.com/wp-content/uploads/219772.image4.png)