00:01
We are going to approximate the following integral using the trapezoidal rule.
00:06
And the integral goes from 0 .6 to 1 .6.
00:11
And the integral functions x squared times natural logarithm of x plus 1.
00:17
So we are going to define that function f of x as x squared times the natural logarithm of x plus 1.
00:28
And we consider that function defined on the closed interval 0 .61 .6, which is the interval of integration.
00:44
And on that interval, we know that this function is continuous, so the interval exists.
00:51
And to use the trappist -rodler rule, we need to define several things first.
00:58
First, we're going to calculate the step size, which is just the length of any of the intervals we consider to subdivide the interval of integration 0616.
01:14
And to do that, we consider n equal space, or of the same length, i mean, subintervals to divide or to partition 0, 6, 1, 6.
01:50
So step size will be that equal length of any of the n subintervals, and then that length is the length of the interval of integration 1 .6 minus 0 .6.
02:04
That is the upper limit of integration minus the lower limit divided by the number of subintervals, which is big n.
02:13
So h is 1 over n.
02:18
That's step size or length of any of the n equal space of intervals.
02:25
And that n is of course a natural number, that is a positive integer.
02:30
So knowing that, we can define the nodes which determine the subintervals.
02:37
And these are given by xi equal the left -hand point of the integral of integration 0 .6 plus i times h where i is a positive and non -negative integer, the index, which ranges from 0 up to when we have i equals 0 we get x0 equals 0 .6 is just the left -hand point of the integral of integration that's the first node and the last node xn i equal n here we get nh and give it that h is 1 over n hn is 1 so we get 0 .66 plus 1 is 1 .6 which is just the right -hand point of the the interval of integration.
03:36
So the first node is the left -hand point of the interval of integration, and the last node is the right -hand point of the interval of integration.
03:44
And in between, we have nodes, x1, x2, up to xn minus one, and each pair of consecutive nodes define a subinterval of length one over n...