5.1 Approximating Areas

[STARTS AT 3:14]

PROFESSOR: The sum of the terms i minus 3 cubed for i equals 1, 2, 3, all the way to 200. So this is starting at i equals 1, going to 200 of i minus 3 squared. So i equals 1 to 200. That would be i squared minus 6i plus 9. 

And I'm going to break this up into three summations, break that into i squared minus 6 times i equals 1 to 200 i plus i equals 1 to 200 of 9. I broke that into three different summations. 

Well, the first, the i squared, is going to be n, which is 200, times n plus 1, 201, times 2n plus 1, which will be 401, divided by 6 minus 6 times the sum of i, which is going to be, based on that formula, n, so 200, times 201-- that's n plus 1-- over 2 plus 9 times 200 because, when we're summing up a constant, it's the constant times n. So in this case, it's going to be 9 times 200. All right, and that total is 2,567,900. 

Let's do something very similar for part B, the sum of the terms i cubed minus i squared for i equals 1 through 6. So i cubed minus i squared, i equals 1 to 6. Well, that will be equals 1 to 6 of i cubed, i equals 1 to 6 of i squared. And i cubed is going to be 6 squared. And that'll be 6 squared times 7 squared, n and n plus 1, over 4 minus-- for i squared, we have n, n plus 1, 2n plus 1, which would you be 13, over 6. And that total is 350.

[ENDS AT 6:17]