Integral calculus

Would you believe me if I told you that if you walked straight at a wall that you would never actually get to the wall? Integral calculus allows you to mathematically prove this crazy idea. When you think of calculus, think tiny as in infinitesimal. By subdividing the space between you and the wall into ever smaller divisions, you can mathematically establish that there is an infinite number of divisions, and you can never actually get to the wall. Do not try this at home kids, not without some help from integrals and derivatives, the basic tools of calculus. The study of integral calculus includes: integrals and their inverse, differentials, derivatives, anti-derivatives, and approximating the area of curvilinear regions.
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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 anti-derivative through the Fundamental Theorem of Calculus (which is why we also use the integration symbol for anti-derivatives as well). 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

Sequences, series and approximating functions. Maclaurin and Taylor series.

AP Calculus practice questions

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

Sequences, series, and function approximation

Sequences, series and approximating functions. Maclaurin and Taylor series.
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All content in “Sequences, series, and function approximation”

Estimating infinite series

We've spent a lot of time thinking about whether a series converges or diverges. But, even if we can determine that a series converges, how can we figure out what it converges to? This tutorial will show techniques of estimating what a series converges to and also determining how good our estimates are. This is super useful because most series can't be precisely evaluated (like we were able to do with infinite geometric series).

Power series function representation using algebra

Now that we're familiar with the idea of an infinite series, we can now think about functions that are defined using infinite series. In particular, we'll begin to look at the power series representation of a function (and the special case of a geometric series). In later tutorials, we'll use calculus to find the power series of more types of functions.

Maclaurin series and Euler's identity

In this tutorial, we will learn to approximate differentiable functions with polynomials. Beyond just being super cool, this can be useful for approximating functions so that they are easier to calculate, differentiate or integrate. So whether you will have to write simulations or become a bond trader (bond traders use polynomial approximation to estimate changes in bond prices given interest rate changes and vice versa), this tutorial could be fun. If that isn't motivation enough, we also come up with one of the most epic and powerful conclusions in all of mathematics in this tutorial: Euler's identity.