00:01
In this problem, we are given this function f of x equal to 3x squared on the interval from 5 to 8.
00:10
And we are going to compute the area under the graph of this function using a riemann sum that involves some arbitrary number of subintervals.
00:24
So we have s n equal to summation from k equal to 1 to n, a k.
00:32
So here s n is the sum of all the areas of these rectangles, which there are n of them.
00:41
And a k is the area of the kth rectangle.
00:46
So we are going to find this sum and then we are going to take the limit as n goes to infinity.
00:53
Of course, along the way, we are going to compute some certain quantities and i'm going to describe them as we go.
01:02
Now the area for this kth rectangle can be written as this function evaluated at the kth point on this interval times delta x, where delta x is the uniform width for these rectangles.
01:22
So we have 5 here, 8 here and we want n subintervals.
01:29
So each step is delta x.
01:34
And if we have n subintervals, then delta x is given by 8 minus 5 over n, which is 3 over n.
01:47
So this is the first question, namely the length of each subinterval.
01:54
Then we have, let's say somewhere here, the kth point.
02:01
So xk is given by the initial point plus k times delta x.
02:09
So we have 5 plus 3k over n.
02:15
And this is the answer to the second question, namely a formula for the kth division point in terms of k and n.
02:25
Okay, now let us write down the kth area.
02:29
So ak is equal to f of xk times delta x.
02:34
So we have 3xk squared times delta x.
02:40
So we have 3 times 5 plus 3k over n squared delta x.
02:48
Delta x is 3 over n.
02:51
And if we expand this expression, we obtain 81 over n cubed k squared plus 270 over n squared k plus 225 over n...