Author: Ushanka

This is the equation for a normal distribution curve, one of the more important functions in probability and statistics:

It is the very recognizable bell-curve shape.

In the equation above, μ is the expected value of the distribution. It also corresponds to the point where the function has the greatest y-value. σ is the value of one standard deviation. Ask...ask your stats instructor. When μ=0 and σ=1, we get what is called the standard normal distribution curve:

The area under this standard normal distribution curve is equal to exactly 1. I am going to show you how to integrate this, because I think it is the coolest integral I have ever seen.

Our limits of integration are plus and minus infinity. We'll say the area under the SNDC is A.

Oh, geez, that is a mean ol' integral. That would be a real pain to solve. Sure, I guess Wolfram Alpha can do it, but that is stupid and lame and stupidly lame. We're going to do this the cool person's way. You do want to be a cool person, right? Right. Now, let's note something obvious: the SNDC in the x-y plane will have the same area A as the SNDC in the x-z plane, but the SNDC will have the equations

instead. Similarly, the SNDC in the x-z plane will have the same area A as the SNDC in the y-z plane, but it will have a different variable of integration.

Now, since we've got two integrals that both equal A, we can multiply them together to get a double integral that equals A^{2}.

This gives us a three-dimensional bell curve, a surface in space that looks like this:

(Apologies for the two missing latitude/longitude lines. I'm not exactly sure what happened to them, but I suspect they escaped my picture and fled to Venezuela.) We can observe from the visual representation of the surface that this is a thing which could be represented in polar coordinates. Here's a quick refresher of Important Polar Coordinate Facts.

I guess the second of those Important Facts is unnecessary for this integral, but it is good to know nonetheless. So, when we make the appropriate substitutions, we get:

Since there's no θ in the formula itself, that one is pretty easy to integrate.

Now this is a nice integral. We can use substitution and quickly get the following:

I assume you're good enough with the limits and the second part of the Fundamental Theorem of Calculus to follow my logic in this next step.

I assume you're good enough with arithmetic and spatial reasoning to follow my logic in this next step.

If you try this again with a (nonstandard) normal distribution curve, you will see that the area is still 1. Yep, try it and see!

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