00:01
We are being asked to estimate the following integral using the simpsom's one -third rule.
00:06
One -third rules means we need to divide the interval into three parts.
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So if we draw a number line, our initial part is at zero, our final part is at pi over two.
00:20
So to divide it into three different segments, not three parts, like three different values.
00:25
We have the initial, we have the final, then we need just the middle part.
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And the middle part between 0 and pi over 2, we can get it by dividing pi over 2 over 2 is equal to pi over 4.
00:38
So these are the three values we're going to use for the simpson's rule.
00:43
For simpson's rule, we need to do the integral or the area, let's put it as area, is equal to b minus a, where b is the upper limit, a is the lower limit.
00:56
Of f -fx 0 plus 4 times f -fx 1 plus f -fx 1 plus f -fx and this will be divided by 6 this is the rule of this is the formula for simpson's rule using the 1 -third so dividing the interval into three different values so b minus a b is just pi over 2 minus a this is 0 now we need to do f of x 0 is 0 so what we do is we replace 8 plus 4 cosine instead of x we write cosine of 0 plus 4 sorry yes 4 times 8 plus 4 times cosine of pi over 4 which is the second part this is not a nice 4 let's fix it so we can have it better in pi over four.
02:03
There we go.
02:05
And we have plus.
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The last part is cosine.
02:11
Sorry, first we go with the eight plus four times cosine of pi over two.
02:20
All of this is going to be divided by six.
02:25
Now we can simplify some parts here, like, for example, cosine of pi over two is zero.
02:31
So this part is gone, leaving us with just 8, then cosine of 0 is 1.
02:37
So it's just 8 plus 4 plus 8.
02:40
And then, i mean, we can do some simplifications here, or since we can use calculators, let's put it up the calculator, and we get that the answer is equal.
02:49
So the area under this integral, using simpson's rule, is 16 .576.
02:58
This is the area using simpson's rule...