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 a, b, let me write that, 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 gonna get the area of a rectangle. The rectangle would be the area of this rectangle right over here, and that rectangle is 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 could 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 of this to come up 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.