00:02
Okay, in this question, let's read it first.
00:06
Use trapezoidal and simpsons in numerical integration to find two estimates of natural log of 2 by approximating the integral dx using partition with 8 subvenevales, provide bounds on the errors of the estimates.
00:25
So let's begin.
00:26
The first point here is to write out how we're going to work.
00:33
Generally, to find the trapezoidal approximation of an integral, we do this delta x over 2 times plus 2 f plus 2 f of x to and on and on up to f of x.
00:58
So we're going to find firstly this little guide over here so let's do it we can write the following notice the upper the upper bound is two and the lower one is one and we're going to divide by n in this case we got eight sub -intervolveld so we can write and our delta x is equal to one over 8 but notice that we only average of delta x so delta x so delta x over 2 is going to be 1 over 16 so here we got the first part the third part we're going to find the sub intervals since it's a trapezoid we're going to count you're going to add 1 plus 1 over 8 is going to be 9 over 8.
02:33
Okay, notice that we got 8.
02:40
For each one of the 8 sub -intervals, we got these numbers.
02:47
And these are, those are the ones we're going to plug into our formula.
02:55
We're going to write this trapezoidal approximation, b over a, f of x, this is going to be approximately plus 2 1 over 5 over 4 plus 2 the expression this is going to to be this is approximately as 1 over 16 times 1 plus 16 over 9 plus 8 over 5 look at that now this is going to be approximately 0 .6941.
04:26
If we now let's take the let's integrate the when we integrate 1 over x the x this is going to be 0 .69 3147 so the error let's do it 69, 4, 1, 2, and the approximation.
04:58
This is the approximation.
05:00
All right.
05:01
Now let's go to the second part of the question.
05:05
That is to find the approximation and approximation for when we integrate upper bound to lower bound 1, 1 over x, d.
05:16
This is approximately equal to dealing with the simpson's rule...