Worked example finding area under density curves | AP Statistics | Khan Academy

Consider the density curve below. This density curve doesn't look like the ones we typically see that are a little bit curvier, but this is a little easier for us to work with and figure out areas.

They ask us to find the percent of the area under the density curve where x is more than two. So, what area represents when x is more than two? This is when x is equal to two, so they're talking about this area right over here.

We need to figure out the percent of the total area under the curve that this blue area actually represents. So first, let's find the total area under the density curve. The density only has area from x equals one to x equals three.

So, it does amount to finding the area of this larger trapezoid. Let me highlight this trapezoid in red. We want to find the area of this trapezoid right over here, and that should be equal to 1 because all density curves have an area of 1 under the total curve.

Let's first verify that. There are a couple of ways to think about it. We could split it up into two shapes, or you could just use the formula for the area of a trapezoid. In fact, let's use the formula for the area of a trapezoid.

The formula for the area of a trapezoid is you would take the average of this length. We do that in another color. This length and this length, let's see, this is 0.25. 0.25 plus this height, 0.75, divided by 2. So that's the average of those two sides times the base.

The average of this length and this length, let's see, this is 0.25 plus 0.75, which is equal to 1, so the area under the entire density curve is 1, which needs to be true for this to be a density curve.

Now, let's think about what percentage of that area is represented in blue right over here. Well, we could do the same thing. We could say, "All right, this is a trapezoid." We want to take the average of this side and this side and multiply it times the base.

This side is 0.5 high, 0.5 plus 0.75, 0.75 high, and we're going to take the average of that divided by 2 times the base. The base going from 2 to 3 is only equal to 1. So, times 1, and this is going to give us 1.25 over 2.

What is that going to be equal to? Well, that would be the same thing as zero point... What, let's see, 0.625. Did I do that right? Yep. If I multiply 2 times this, I would get 1.25.

So, the percent of the area under the density curve where x is more than 2—this is the decimal expression of it—but if we wanted to write it as a percent, it would be 62.5.

Let's do another example. Consider the density curve below. Alright, we have another one of these somewhat angular density curves. Find the percent of the area under the density curve where x is more than three.

So, we're talking about... see, this is where x is equal to three. x is more than three; we’re talking about this entire area right over here. Well, this is actually somewhat straightforward because if we're saying the area where x is more than three, that's the entire area under the density curve.

The entire area under any density curve needs to be equal to one. Or you could say, "Find the percent of the area under the density curve." Well, the whole density curve is where x is more than three, so one hundred percent. We don't even have to go through the trouble of trying to directly calculate the area.