A secret of the definite integral

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Two young mathematicians discuss what calculus is all about.

Check out this dialogue between two calculus students (based on a true story):

Devyn: Ah. So now we have a connection between derivatives and integrals.
Riley: Right, the derivative of the accumulation function is the ‘‘inside’’ function.
Devyn: So how do we use this to compute area?

Sometimes it helps to think about the most basic examples. Consider $\int _2^5 4 \d t$ We know (by geometry) that this computes the area of a $3\times 4$ rectangle which equals $12$ . On the other hand, if we consider the accumulation function $F(x) = \int _2^x 4 \d t,$ we see that $F(5) = \int _2^5 4 \d t.$

What is $F(2)$ ? $F(2) = \answer {0}$

On the other hand, the First Fundamental Theorem of Calculus says that if $F(x) = \int _2^x 4 \d t,$ then $F'(x) = 4$ . Armed with this knowledge, and the fact that $F(2) = 0$ , what must $F(x)$ be? $F(x) = \answer {4x-8}$