I know how to integrate, but I still never see any real world example of it, so it is difficult to understand. Why is it useful?

(In tandem with the whiteboard) Integration is a way of working out the total of something. An example of this is if you see a graph of velocity and time, the area under the graph is the total distance travelled. This makes sense if the velocity is constant, you don't need a complex mathematical process to see what the distance is. But how about if the velocity isn't constant?

Let's look at a simple example, which uses a little bit of physics. Let's examine the total force that water exerts on a wall. This will demonstrate the way to derive an indefinite integral from first principles and then apply it - it should then be more apparent what the purpose of integration is.