# Lesson Series

Integral Calculus is the study of areas under curves and their application to real world problems. The area under a curve between points $a$ and $b$ are denoted by \begin{equation}\mbox{Area between }a\mbox{ and }b=\int_a^bf(x)\,dx.\end{equation}
We discuss definite and indefinite integrals and take a quick look at the Fundamental Theorem of Integral Calculus.

We divide the $x$-axis between the limits of integration into an arbitrary set of closed intervals. We provide notation and use pictures to show several arbitrary partitions.

We define the upper and lower sums of a bounded function and a partition on it. From the upper and lower sums we define the definite integral and provide examples.

In this lesson we define the term partition limit which we denote by
\[
\lim_{||P\,||\to 0}F(P)=L.
\]

In this lesson we are going to go through the logical framework that defines a definite integral. The main goal of this lesson is to show that when \[\lim_{||P\,||\to 0}\,L(P)\] and \[\lim_{||P\,||\to 0}\,U(P)\] are equal, then they are equal to \[\int_a^bf(x)\,dx.\] We also provide three equivalent definitions of a definite integral.

We define the definite integral as the limit of Riemann sums.

Here we describe how a definite integral and the area it represents can be negative. There are typically two ways this can happen. The first is when the function $f(x)$ is negative. The second is when we integrate in the opposite direction so that
\begin{equation}
\int_a^b\,f(x)dx=-\int_b^a\,f(x)dx.
\end{equation}

The Fundamental Theorem of Integral Calculus is the main technique we use to determine the value of the definite integral $\int_a^b\,f(x)dx$. The theorem states that
\[
\int_a^b\,f(x)dx=F(b)-F(a)
\]
where $F(x)$ is the antiderivative of $f(x)$.

In this lesson we provide proofs of some of the most common properties of integrals that every student should know.