00:03
So problem 23, here we're asked to use the limit definition of a definite integral to figure out the exact value of this integral.
00:12
So the definition is going to be the limit as n approaches infinity of the sum of i equals 1 to n, and you're summing up the area of rectangles, in rectangles, but then letting n go to infinity.
00:30
So the width of each rectangle is going to be 0 minus minus 2 over n, which is going to be 2 over n.
00:40
So that is the width of each rectangle in this case.
00:44
And the height of each rectangle is going to be determined by stepping along from 1 to n and evaluating this function.
00:51
So it's going to be, if you think about it, so x squared is the function.
00:56
So you're going to have negative 2 is the boundary.
00:59
So plus and then you step each increment is 2 over n.
01:05
So 2i over n squared plus negative 2 plus 2i over n.
01:17
And then the width of each rectangle, as we determined, was 2 over n.
01:28
So what i want to do, the easiest thing to do is to break this up into two integrals...