3. Utilize the number of divisions n=20 to evaluate the...

3. Utilize the number of divisions $n=20$ to evaluate the integral $\int_0^{\pi/4} \ln(2 + \tan x)dx$ by using (a) trapezoidal rule, (b) Simpson's one-third rule, and (c) five-point Gauss-Legendre quadrature formula, respectively.

Transcript

00:01 In this problem, we're asked to use the trapezoid rule and simpsons rule to approximate a definite interval.

00:08 So i've written out the trapezoid rule and simpsons rule here.

00:11 I'm going to start with trapezoid.

00:13 So first thing you'll notice is we're going to need to solve for delta x.

00:21 So let's go ahead and scroll down.

00:23 Take a look.

00:24 I've written out the definite integral here.

00:27 And then our f of s, which is the equation inside that, as well as our auxux.

00:33 A, which is our starting point, b, which is our ending point.

00:38 And then finally, we have 20 sub intervals, which the number of sub intervals is sometimes 10.

00:46 So keeping that in mind, delta x equals b minus a over n.

00:54 So b in this case is 5, a is 4, and n is 20.

00:58 So we get our delta x equal down 20, for a chance decimal 1 0 .05.

01:06 So now we can actually apply.

01:10 The trapsoid rule.

01:12 So something i want to note is that when we're adding up our intervals here, our first interval is a coefficient of one, and our last interval has a coefficient of one.

01:25 But each subsequent interval in between here is going to have a coefficient of two.

01:31 So we're going to have 20 of these intervals, which is quite a bit.

01:38 I'm not going to write all them out, but you're going to have to plug them all into your calculator.

01:43 But i'll kind of write out the outline here.

01:53 So again, if you recall, we have the coefficient that we're multiplying all of these guys by is delta x over 2.

02:05 So we just solved for delta x, that was 1 .20th.

02:10 And then we're dividing this by 2.

02:13 All right, and then for the rest of our terms, again, we start with f of x not, which has a coefficient of 1 in the front there.

02:23 What is x9? well, x9 is just going to be our starting point, which is 4.

02:28 So this is f of 4.

02:32 But what is f of 4? well, we identified the equation of the interval.

02:40 We're actually trying to solve 4.

02:43 And f of 4 is just going to be 3 plus 4 to the 6th.

03:05 So the next sub -interval is going to be 2.

03:10 Again, each subsequent interval is going to have a coefficient of 2 times 3, until we get to the end, times 3 times 4 .05 to the 6.

03:19 So where did i get this number? well, that's just x0 plus delta x, right? and we know delta x equals 120th or 0 .0 .5.

03:31 All right.

03:38 So the next interval is going to be 2 times 3 plus adding 0 .05, 4 .10.

03:46 Sorry that's to the sixth not squared and we're going to keep doing this all the way so we'll get to our last interval with coefficient of two which is three plus 4 .95 to the sixth and finally again the very last interval it has a coefficient of one like our first interval and the last interval if you do it right should be five which makes sense so that is our ending point...

Question

3. Utilize the number of divisions $n=20$ to evaluate the integral $\int_0^{\pi/4} \ln(2 + \tan x)dx$ by using (a) trapezoidal rule, (b) Simpson's one-third rule, and (c) five-point Gauss-Legendre quadrature formula, respectively.

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