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Definite Integrals and Area - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Definite integrals can be used to find the area under, over, or between curves. If a function is strictly positive, the area between it and the x axis is simply the definite integral. If it is simply negative, the area is -1 times the definite integral. If finding the area between two positive functions, the area is the definite integral of the higher function minus the lower function, or the definite integral of (f(x)-g(x)).

I want to talk about how definite integrals can be used to find area. First of all there are 2 basic kinds of area problems. First the area between y=f of x some curve and the x axis from x=a to x=b. That situation looks like this.
So if this is your graph of y=f of x. If your graph, if your function's entirely not negative on this interval that is it's above the x axis, then the definite integral will give you the area exactly. However, if your function y=f of x is down here, if it's below the x axis, if it's non-positive, the definite integral does not give you the area it gives you the opposite of the area. So if you're looking for this area, you can use the definite integral but you just have to remember to flip the sign. You're going to get a negative number, you should make it positive. So, just keep that in mind. When the function's negative, you'll get the opposite of the area, when it's positive you'll get the area.
And then the second problem is the area between 2 curves. so let's say our 2 curves are y=f of x and y=g of x. And we're interested in the area between x=a and x=b and here's the situation. y=f of x is the higher of the 2 curves, y=g of x is the lower. So that's what I've said here. When f of x is greater than or equal to g of x on the interval the area, this area here, this band is going to be equal to the area under y=f of x. That's all the area under this curve and above the x axis minus the area under y=g of x. That's this area here. So take the whole area, subtract this away. And each of those areas can be represented by integrals. So this is the area under f of x. This is the area under g of x and we subtract because it turns out that you can write this difference of integrals as the integral of the difference of the functions. So you can inte- so in one integral you can get the entire area between 2 curves. It's the integral from a to b, from left end point to right end point of top function minus bottom function. Now we'll use this in upcoming problems.

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