# 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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# 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!
All content in “Integration applications”

## 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. This tutorial focuses on the "disc method" and the "washer method" for these types of problems.

## 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.

## Solid of revolution volume

Using definite integration, we know how to find the area under a curve. But what about the volume of the 3-D shape generated by rotating a section of the curve about one of the axes (or any horizontal or vertical line for that matter). This in an older tutorial that is now covered in other tutorials. This tutorial will give you a powerful tool and stretch your powers of 3-D visualization!

## Area defined by polar graphs

We'll now use the power of the integral to find areas defined by polar graphs!

## 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!