# Integrals

Contents

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.

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.

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.

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.

As with any other mathematical operation, studying the properties of definite integrals will be a way of kicking the tires and getting a feel for how they work. Not only will this sharpen your understanding of integrals, but it will help to make computations easier down the road.

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

The very powerful and very surprising fact at the heart of calculus is that integrals and derivatives are, in a sense, opposite to one and other. At first glance, rates of change and slope might seem completely unrelated to the area under a curve. In this tutorial, you will learn about the aptly named "Fundamental theorem of calculus".

Here we'll see how the fundamental theorem of calculus lets us compute definite integrals using antiderivatives. In some sense, this is what we have been building up to for the last few tutorials.

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".