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Worked example finding area under density curves
- [Instructor] Consider the density curve below and 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. So 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. And so 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, the density curve 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. So we want to find the area of this trapezoid right over here. And then that should be equal to one because all density curves have an area of one under the total curve. So let's first verify that. There's a couple of ways to think about it. We could split it up into two shapes or you could just use the formula for an area of a trapezoid. In fact let's use the formula for an area of a trapezoid. The formula for an area of a trapezoid is you would take the average of, you would take the average of this length. Let me do that in another color. This length and this length and then multiply that times the base. So the average of this length and this length, let's see this is 0.25. 0.25 plus this height, 0.75, divided by two. So that's the average of those two sides times the base, times this right over here which is two. And so this is going to give us as it should have, 0.25 plus 0.75 which is equal to one. So the area under the entire density curve is one which we need 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. So 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 two times the base. Well the base going from two to three is only equal to, is equal to one. So times one. And so this is going to give us 1.25, 1.25 over two. And 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 two times this, I would get 1.25. So the percent of the area under the density curve where x is more than two, this is the decimal expression of it. If we wanted to write it as a percent, it would Be 62.5% Let's do another example. Consider the density curve below. All right 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, let's 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. And 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 100%, we don't even have to go through the trouble of trying to directly calculate the area.