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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Indefinite integrals

If f is the derivative of F, then F is an antiderivative of f. We also call F the "indefinite integral" of f. In other words, indefinite integrals and antiderivatives are, essentially, reverse derivatives. Why differentiate in reverse? Good question! Keep going and you'll find out!

Definite integrals introduction

Definite integrals are a way to describe the area under a curve. Make introduction with this intriguing concept, along with its elaborate notation and various properties.

Riemann sums

Riemann sums is the name of a family of methods we can use to approximate the area under a curve. Through Riemann sums we come up with a formal definition for the definite integral.

Fundamental theorem of calculus

So you've learned about indefinite integrals and you've learned about definite integrals. Have you wondered what's the connection between these two concepts? You will get all the answers right here. Beware, this is pretty mind-blowing.

Definite integral evaluation

Make your first steps in evaluating definite integrals, armed with the Fundamental theorem of calculus.

Integration techniques

Some functions don't make it easy to find their integrals, but we are not ones to give up so fast! Learn some advanced tools for integrating the more troublesome functions.

Area & arc length using calculus

Become a professional area-under-curve finder! You will also learn here how integrals can be used to find lengths of curves. The tools of calculus are so versatile!

Integration applications

As with derivatives, solve some real world problems and mathematical problems using the power of integral calculus.

Volume using calculus

Integrals can be used to find 2D measures (area) and 1D measures (lengths). But it can also be used to find 3D measures (volume)! Learn all about it here.

Sequences

You may have learned about sequences before. Sequences serve as the foundation for series, which are really important in integral calculus. Review you knowledge of sequences right here.

Series

Series are simply suns of terms in sequences. This simple innovation uncovers a world of fascinating functions and behavior.

Power series

Power series are infinite series of the form Σaₙxⁿ (where n is a positive integer). Even though this family of series has a surprisingly simple behavior, it can be used to approximate very elaborate functions.
Integral calculus: Questions

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