Integral calculus

How do you find the area under a curve? What about the length of any curve? Is there a way to make sense out of the idea of adding infinitely many infinitely small things? Integral calculus gives us the tools to answer these questions and many more. Surprisingly, these questions are related to the derivative, and in some sense, the answer to each one is the opposite of the derivative.
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Integrals

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.

Integration techniques

We know that a definite integral can represent area and we've seen how this is connected to the idea of an antiderivative through the Fundamental Theorem of Calculus. Unfortunately, integrals aren't always easy to compute. Now, we'll build out our toolkit for evaluating integrals, both definite and indefinite!

Integration applications

Let's now use our significant arsenal of integration techniques to tackles a wide variety of problems that can be solved through integration!

Sequences, series, and function approximation

Now we switch gears away from integration to talk about sequences and series. Much of calculus is about dealing with infinity, and this topic has us dancing very closely with infinity itself. Sequences are infinite lists, series are infinite sums, and there is no small amount of delicacy involved in managing these two objects.

AP Calculus practice questions

Sample questions from the A.P. Calculus AB and BC exams (both multiple choice and free answer).

Integrals

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.
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All content in “Integrals”

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.

Properties of the definite integral

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.

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.

Fundamental theorem of calculus

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".

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".