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Studying for a test? Prepare with these 9 lessons on Antiderivatives and the fundamental theorem of calculus.
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Video transcript
-[Voiceover] What I want to do with this video is think about the idea of an average value of a function over sum closed in full. So what do I mean by that, and how can we think about what average value of a function even means? So let's say that's my Y axis, and let's say that this is my, this right over here, is my X axis, and let me draw a function here. So let's say the function looks something like that. That's the graph of Y is equal to, Y is equal to F of X. And I always think about a closed interval. So we're going to think about the closed interval between A and B, including A and B, that's what makes it closed, we're including our endpoints. So we're going to think about this interval right over here. So between X is equal to A, and X is equal to B, what is the average value of this function? One way to think about it is, what is the average height of this function? So how could we, what would that mean? Well, one way to think about it, it would be some height, so that if we mutiplied times the width of this interval, we'll get the area under the curve. So, another way to think about it is, the area under the curve right over here, the area, just let me do this in a different color, so the area under this curve right over here, I'll shade it in yellow, we already know that we can express this as the definite integral from A to B, of F of X, DX. The average value of our function over this closed interval AB, let me write that, over, over the closed interval between A and B, including A and B, we could think about it as some height, some height, let me do this in a new color, some value of our function, some height, let me think about it, maybe some height right over here, so that if we multiply this height times this width, we're going to get the area of a rectangle. The rectangle would be the area of this rectangle right over here, and that rectangle's going to have the same area as the area under the curve, which is a reasonable way, if you kind of remember how when you even think about finding the area, or one way to think about the area of a trapezoid, you can multiply if you have a trapezoid. If you have a trapezoid like this, you have a trapezoid like this, this is, you can kind of turn 90 degrees, but you multiply the height times the average width of the trapezoid, and then that will give you its area, so this would be the average width, which in a trapezoid like this would just be halfway between. This function is not linear, so it's not necessarily going to be halfway in between, but it's that same idea. So how could we use this idea, where this is right over here, this height, this height right over here, we could call this, we could call this the function's average, the function's average. How could we use all this to come back with a formula for the average of a function over this closed interval? Well, let's just express in math what we've already said. We already said that this function average should be some height, so let's say the function average, so that's a height, and if I multiply it times the width of this interval, so this width right over here, this width right over here is just going to be the larger value minus the smaller value, so it's going to be B minus A, so the average value of the function times the width of the interval should give us an area that is equivalent to the area under the curve. So it should be equal to the definite integral from A to B of F of X, DX. And so if we just, if we knew all of this other stuff we could solve for the function's average. The function's average, we divide both sides by B minus A, the function's average is going to be equal to, just dividing both sides by B minus A, you're going to get one over B minus A, times the definite integral, the definite integral from A to B of F of X, DX. Or another way to think about it, you're going to figure out what the area under the curve is over that interval, you're going to divide that by the width, and then you're going to have the function's average. One way to think about it, you're going to have the average height. And once again, I'd like to remind you that, because you shouldn't just sit there and try to memorize this thing, just get a conceptual understanding of what this is really just trying to say. Area under the curve divided by the width, well that's just going to give you the average height, that's going to give you the average of the function. In the next video, we'll actually apply this formula, you'll see that it's actually straightforward to calculate if you can figure out the definite integral.