00:01
For this question, we want to approximate the definite integral using trapezoidal rule and simpson's rule with n equals 4.
00:06
And then we want to use a graphing utility to get its approximate.
00:11
So the integral is from 0 to pi, the square root of pi over 4, the tangent of x squared the x.
00:34
So using the trapezoidal rule.
00:38
And remember the trapezoidal rule is we're just adding areas of trapezoids.
00:42
So we have the one -half times base 1 plus base 2 times the height.
00:47
And the one -half and the height stays constant.
00:50
So we do can factor that out to the front.
00:54
So we have one -half times the height.
00:57
The height of the trapezoids is, remember, the difference of a and b divided by n.
01:05
So that's going to be the square root of pi over four minus zero, which is just that over four.
01:17
And then remember we have all of the function values which give us the base.
01:22
So the first one is going to be f of zero plus.
01:29
And then all the ones in the middle are two times because each one's in the middle are used as the base for two different trapezoids.
01:37
So two f of, and here's the width.
01:40
So this is what we're going to be adding.
01:42
So f of, well, the first time we started at zero, and if we add this, then we'll just be at f of the square root of pi over four, over four, plus two f of.
02:06
I'm just going to add that again.
02:07
So i'm going to have that same thing over two times that same thing over four, so two fours.
02:12
So that's going to be one half.
02:13
So i'm going to get the square root of pi.
02:18
Over 4 over 2 plus 2 times f of adding it again i'm going to end up with 3 4s so 3 square root of pi over 4 over 4 and then lastly just one of these f of 4 4 4 or just the square root of pi over 4 all right so let's go ahead and start to simplify that mess so i guess it is need to quite do that yet.
03:04
So let's get this.
03:09
This is going to be square read of pi over 4 over 8 times f of 0.
03:21
If i substitute a 0 and for x, i get x squared is 0 or 0 and the tangent of 0 is 0.
03:29
So it's going to be 0 plus 2 times the tangent of x squared...
