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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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!
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All content in “Integration applications”

Area between curves

By integrating the difference of two functions, you can find the area between them.

Average value of a function

We don't need calculus to figure out the average value of a linear function over an interval, but what about non-linear functions? Luckily, integral calculus comes to the rescue here. In this tutorial, we'll understand what "average value" of a function over an interval means. We'll also connect that notion to the Mean Value Theorem we first learned in differential calculus.

Arc length

We'll now use integration to find the arc length of a curve. As we'll see, it is based on the same idea of summing up an infinite number of infinitely small line segments.

Volume of solids with known cross sections

We will now leverage the definite integral to find volumes of figures where we know what the cross sections look like. It is surprisingly fun.

Solids of revolution - disc method

You know how to use definite integrals to find areas under curves. We now take that idea for "spin" by thinking about the volumes of things created when you rotate functions around various lines.

Solids of revolution - shell method

You want to rotate a function around a vertical line, but do all your integrating in terms of x and f(x), then the shell method is your new friend. It is similarly fantastic when you want to rotate around a horizontal line but integrate in terms of y.

Area defined by polar graphs

There's no reason to limit ourselves to cartesian coordinates. When a curve is defined with polar coordinates, and we want to find the area between the curve and the origin, so to speak, we use the method taught in this tutorial.

Arc length of polar graphs

You may already be familiar with finding arc length of graphs that are defined in terms of rectangular coordinates. We'll now extend our knowledge of arc length to include polar graphs.