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!

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

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.

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.

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.

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.

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.

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.