>> There are two classic problems in calculus. One of them is the tangent line problem which we've already considered, and the other is the area problem which we are about to consider. Now, the tangent line problem was related to the idea of derivative and we developed that using the notion of limits. Now, we're gonna talk about the area problem, the other classic problem in calculus here in just a little bit. And it's going to be developed in terms of limits as well. And one of the ideas that are part of this structure, part of this development is the idea of summation. And we're gonna begin the development of the area problem by talking about the notion of summation. Summation has a certain notation associated with it. This is the Greek letter, sigma. And it stands for summation and "I" is called the "index of the summation." 1 is the lower bound and N is the upper bound of the summation. And this is read, this is the sum as I runs from 1 to N of terms that are in the form A subscript I, A subscript I. Now, I is a counting mechanism. So what this means is that I takes on values from 1 to N. So, the first item here would be A subscript 1 and then there would be A subscript 2, A subscript 3 and so on all the way through to A subscript N. Now, [coughs] a practical example would be this, the sum as I runs from 1 to 5 of 2I would be 2 times 1. You see, we're starting at the lower bound and we're going to the upper bound. So it would be 2 times 1, plus 2 times 2, plus 2 times 3, plus 2 times 4, plus 2 times 5. Now we can actually evaluate it. It's pretty easy for this problem, 2 plus 4 plus 6 plus 8 plus 10, or 30. But the idea is how to perform the expansion of this summation idea. Now, the lower bound doesn't always have to start at 1 nor does the counting mechanism have to be designated I. Here, we have the sum as K runs from 3 to 7 of K squared. So, we'll just replace K with the values from 3 to 7, you see, because K is the counting mechanism and it turns out to be 3 squared plus 4 squared plus 5 squared plus 6 squared plus 7 squared. Okay, so that's the idea involved in summation notation. We're gonna talk about a few nuances here and some formulas having to do with summation notation. Here is the sum as I runs from 1 to 4 of a constant 5. Now, notice that I doesn't appear here in this expression. So, when I is 1, what is the value of the expression? Well, it's 5 and we're gonna add. Now, when I is 2, what is the value of the expression? Oh, it's still 5, you see. And the idea is that the value of the expression, the value of this doesn't change every time the counting mechanism jumps to another number. So it stays the same for each of the values in the count. So when I is 1, when I is 2, when I is 3, and when I is 4, we have the value as 5. Now, the shortcut to this, of course, since we have four 5s, we could say then that this is 4 times 5 or 20 of course. But the general idea is this, that if we have a summation as I runs from 1 to N of some constant C, then it's just the upper bound times that constant to evaluate the sum. And this of course holds true though only if the lower bound is 1.