In this topic, we are going to connect the two big ideas in Calculus: Instantaneous rate and area under a curve. We'll see that a definite integral can be thought of as an infinite sum of infinitely small things and how this connects to the derivative of a function.

Indefinite integral as anti-derivative

You are very familiar with taking the derivative of a function. Now we are going to go the other way around--if I give you a derivative of a function, can you come up with a possible original function. The symbol which we'll use to denote the anti-derivative will see strange at first, but it will all come together in a few tutorials when we see the connection between areas under curves, integrals and anti-derivatives.

Area and net change

Differential calculus was all about rates. That is, after all, what a derivative is. As we'll see, integral calculus is all about the idea of summing or "integrating" infinitely many infinitely small things to get a finite value. This often represents the area under a curve. This tutorial is a preview of the connection between these two concepts.

Riemann sums

In this tutorial, we'll think about how we can find the area under a curve. We'll first approximate this with rectangles (and trapezoids)--generally called Riemann sums.

Functions defined by integrals

Let's explore functions defined by definite integrals. It will hopefully give you a deeper appreciation of what a definite integral represents.

Improper integrals

What if you want to compute the area of a curve over an infinite region? That turns out to be a perfectly fine thing to do. The bounds of your integral might be infinity (or negative infinity), but with a little help from limits you will be able to handle this situation. Even though nothing is "improper" about doing this, integrals with infinite bounds have been given the unfortunate name of "improper integrals".