>> We're ready for a definition for the calculation of area and that definition looks like this, let F be continuous and non negative on a certain interval. And it's non negative because we want the graph to be above the x axis or it could be on it, but non negative means it won't fall below the x axis. We're talking about area and area is considered to always be positive. But at any rate we have a continuous function that's non negative on a particular interval from A to B and the area then of the region bounded by the graph of the function, the x axis and the vertical lines x equals A and x equals B is this. And the area is calculated as the limit as n approaches infinity of the sum of as I runs from 1 to n of F of C sub I, and I'll talk about it in a minute, times Delta x. Now the Delta x is the width of each of the rectangles and that Delta x can be calculated like this if we want equal widths and right now, that's what we're considering just for the ease of calculation. Delta x is the length of the interval. B minus A is just the length of the interval divided by N. Now n is approaching infinity, I realize, but the Delta x can be represented like this in our calculation, in this formula. Now C subscript I is just a point within the I interval. That's all it amounts to. It's kind of like this. We've got a situation where we have a curve and let's say we've got an A and a B identified and so we're looking for the area under the curve between the curve and the x axis and these two end points of the interval. Now we've got a sub interval here. Now I'm making it large just to be able to write on the board and have you see something, but C sub I is just any point within the interval. So this is just C sub I. You see, and that's the I interval. Okay? And F of C sub I would be, if I extend out here, it would be that distance you see, this vertical distance as F C sub I. Okay. And when we make the calculation for this interval, we're finding the area right here, but then when we talk about the infinitude of these, we're talking about the sum of all such things and therefore we get the area under the curve. Now the Delta x is just the width of the interval.