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{\Large\bf Fundamental Theorem of Calculus}

\vspace{.3in}

As the name suggests, the Fundamental Theorem of Calculus (FTC)
is an important theorem. The theorem connects integrals
and derivatives. There are two versions. 

\vspace{.3in}

{\bf Version 1}

\vspace{.2in}

Define

$$
F(t)=\int_{a}^{t}f(x)dx
$$

where f(x) is a continuous function. (This assumption can be weakened.)
In other words, F(t) is simply the area under the f(x) curve from a to
t. The Fundamental Theorem of Calculus states

$$
F'(t)=f(t)
$$

There is an analogous result for indefinite integrals. Let

$$
F(t)=\int f(t)dt
$$

Then

$$
F'(t)=f(t)
$$


\vspace{.3in}

{\bf Version 2}

\vspace{.2in}

The second version of the Fundamental Theorem of Calculus
states that 

$$
F(t)=F(a)+\int_{a}^{t}F'(x)dx
$$

This last formula can also be expressed in terms of an indefinite
integral:

$$
F(t)=\int F'(t)dt+C
$$

where C is a constant. 

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