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. 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 that'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, if 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.