Note: I am not a math/calculus genius, but I do enjoy what it offers to us. Hence, I am not an expert in this topic. My explanation is meant to be extremely simple. The graphs below assume that both the F-Zero cars are going on a straight path without any obstructions, accelerating at their maximum.

What is this that we sometimes see in the movies, TV, or even in our calculus book? This sexy elongated s-shape is called the integral. Like people, the integral can come in many forms:

The indefinite integral

The definite integral

I’m sure you’re going, “Man, it’s hot, but what else can it do?”

Although this is the most simple way of putting it, the integral is used to find the area, the volume of a 3-d object, the mass of an object, or the displacement/distance. There are MUCH more uses, but those are the common uses at the beginning calculus level.
What do these forms mean?

In the indefinite integral , we would be finding the entire area/volume/mass.

In the definite integral , we would be finding the area/volume/mass over a range that goes from a to b.

I’ll give an example.

Let’s take one of the best racing games ever, F-Zero for the SNES. I loved playing with the Golden Fox, due to its rapid acceleration:

See the curve that represents the rate of speed? Well, if we knew the equation that gives that curve, then we can use integration to see how far the car would be going at a particular point of the curve. Let’s compare it to the Fire Stingray:

Let’s say that we want to know how far a car is after speeding from point a (starting) to point b (the halfway point). Points a and b represent time, where a=0, and b is the halfway point of the graph. I say half-way because we don’t exactly know what time the halfway point is.

So, what we do is take the definite integral from point a to b, which is the area under the graphs from a to b.

of the Golden Fox:

of the Fire Stingray

The blue shading represents the displacement, or how far the car has gone. So, at point b, the Golden Fox will gain a set distance (because the line is straight the entire way), while the Fire Stingray is still gaining more distance.

However, at point b, which is further along? Well, let’s combine the graphs together:

The above graph shows that at time point b, The Golden Fox will be further along than the Fire Stingray, because the Golden Fox has a larger area (it has the red + blue area) than the Fire Stingray (just the blue).

However, what if we took the integral from point a (time = 0) to infinity (time = MAX, although that’s not really possible in real life):

In the above graph, I took the (indefinite integral), which means to take the entire area under the curve. So, we have the entire area for the Golden Fox and the Fire Stingray now.

What does it show? At the maximum time, the Fire Stingray is now ahead of the Golden Fox because the area of the Fire Stingray is greater than the Golden Fox. Since both are at their maximum time, the Fire Stingray will eventually outrun the Golden Fox.

However, when we took the of the curves, the Golden Fox was further along at time b, while the Fire Stingray was lagging behind.

Let’s say that the race lasted b seconds. The Golden Fox would win, because the Fire Stingray would never be able to achieve a greater distance until after point b.

I hope you’ve learned a lot today! This was a really simple explanation without going into the actual math of it. Also, feel free to reply and correct me if I’m wrong with any of my stuff.