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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 and so they asked 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 so 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 and the density only has area the density curve only has area from x equals 1 to x equals 3 so it does amount to form 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 1 because all density curves have an area of 1 under the total curve so let's first verify that so 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 so let's use the formula for an area of a trapezoid the formula for an area of a trapezoid is you take the average of you would take the average of this length we 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 2 so that's the average of those two sides times the base times this right over here which is 2 and so this is going to give us as it should have 0.25 plus 0 point seven five 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 alright 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 hi 0.5 plus 0.75 0.75 hi and we're going to take the average of that divided by 2 times the base well the base going from 2 to 3 is only equal to is equal to 1 so x 1 and so this is going to give us one point two five one point two five 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 do I do that right yep if I multiply 2 times this I would get one point two five 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 sixty two point five percent 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 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 that have to control 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

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