Trapezoidal Rule Example 1


Uploaded by TheIntegralCALC on 09.10.2010

Transcript:
Hi everyone. Welcome back to integralcalc.com. We're going to be doing some trapezoidal rule
problems today. The trapezoidal rule is one way to estimate the area under a graph or
under the curve, between the curve and the x axis.
So the problem that we're given is, the integral of dx over 1 plus x squared on the range negative
1 to 2 where n equals 6. So this is the function that we're given.
The trapezoidal rule, we have to remember is from a to b f of x dx equals.
So this left hand side here is representing our function. So they're saying given a function
like this one, which our f of x is, right? We have a range, a to b and then some function
here, then we apply the following formula. It’s f of x0 plus 2 f of x1 plus 2 f x2,
plus dot dot dot, plus 2 f of xn minus 1. I hate formulas like this. They're so ridiculous
but this is the way it's written out. And it looks so intimidating but it's actually
so simple. I just hate it when we write stuff like this and make it way more complicated
than it is. But anyway, this is the formula that we're given and basically all it’s
saying is that... Well, the first thing we need to do is (let's forget the formula for
now) find delta x. So that's the first step. And the way that we find delta x or the change
in x is to use the number of  trapezoids that they have already told us they want us
to use and the range negative 1 to 2. So they've told us that they want us to divide the range
negative 1 to 2 into 6 different trapezoids. So if our graph is like this and here's negative
1 and here's 2 they want us to divide this into six sections. We have 1, 2, 3, 4, 5,
6. So they want us to have (if our graph looks like this or something), they want us to divide
this into these six trapezoids so that we can do something like that to approximate
the area. But anyway, it's actually really simple. All
we do, is to find delta x. We do the top number minus the bottom number 2 minus negative one
and we divide by n, which they've given us. So we've got two minus negative 1 which is
actually 2 plus 1 which is 3 over 6. So delta x is 1/2.
So given that delta x is 1/2, now we need to write down what these values are. That's
like here, here, here, here. All the way through. You need to write down what those values are.
So the first one obviously is negative 1.  The first one is always the bottom number.
The last one is always the top number. But x of 0 will be the first number. Negative
1, this bottom number here. And then we just keep writing, right? x1, since delta x or
the  change in x is 1/2, we add 1/2 to the previous number each time. So this will actually
be negative 1 plus 1/2 is negative 1/2. And then x2 would be 0, we add 1/2 again. x3 would
be 1/2. x4 would be 1. x5 would be 3/2 or one and a half. And x6 would be 2.
We’ve gotten to 2, which is our upper limit here and the last number here. So we know
that we've found all the values of x. This is our range, right? Negative 1 to 2 and we
have a change or delta x of 1/2 each time. So these are our values of x that we're going
to use to plug into this function. See, we have x0, x1, x2, on and on and on, all the
way to xn and it's six and we have x6 here. So that's why we found these values. You just
find delta x then you write down your first value as x0 is the bottom number. You add
delta x each time until you get to the second number or the top number here and then you're
done. You know you have all your x values. So now that we have that, we're gonna go ahead
and apply this formula. Forget about this. You don't need it because that just represents
our function. So just forget about that. This f of x0.
So what this means is here's x0, right? x0 is negative 1. That means we're plugging in
negative 1 to our function here. So we'll plug negative 1 in. So this is the same thing
as 1/1 plus x squared dx, right? dx is just… Since they're multiplied together, they put
dx on the top. But this, we can completely ignore the dx. All we care about is this right
here, 1/1 plus x squared. So our trapezoidal formula here is going to
be negative 1 plugged into this function. So we'll have 1/1 plus negative 1 squared
and then we plug in the second one and then we go all the way through.
So let's see. negative 1/2. We've got negative 1/1 plus negative 1/2 squared (and I hope
that's large enough that you could kind of read it but I plugged in negative 1/2 here).
See though that we have this 2, we've got another 2, got another 2, we have to multiply
this by 2 because the formula has it there. So then we keep going. We've got 2 on the
next one so we do 2 times 1 over 1 plus 0 squared plus 2 times 1 over 1 plus 1/2 squared.
And I'm going to go ahead and (we're running out of room) kind of write on the second line
here. We did 1/2, we need to multiply that by 2, 1 over 1 plus 1 squared, plus 2 times
1 over 1 plus 3/2 squared, plus 1 over 1 plus 2 squared.
Okay. And then we've got delta x. We said delta x was 1/2. We solved for that up there
already. 1/2 over 2. I wanted to kind of just get the formula written
out and then now kind of explain it afterwards. The reason I hate this formula is because
it looks confusing but all it's saying is that you multiply every term, after the first term
and before the last term by 2. So you can see that our first term is not multiplied
by 2 and our last term is not multiplied by 2; every other term in between is multiplied
by 2. And that's all it means. You've got delta x at the end here; you always divide
by 2; that's part of the formula. And then you just go term by term based on what you've
gotten for x which we did, negative 1, negative 1/2, 0, 1/2, 1, all the way up to 2. You just
plug that into your function here and you multiply all the terms in the middle by 2.
Not the first one, not the last one. And that's all it means. Even though this formula looks
really stupid, it's actully easier than you think.
So now that we've done this, let's go ahead and  simplify. So the first term, right?
We've got 1 over negative 1 squared is just 1 so 1 plus 1 on the bottom there is 2, so
we get a 2 on the bottom and then we have 2 times negative 1/2 squared is positive 1/4
plus 1 is 5/4 so that's actually going to be 4/5 because 1 divided by a fraction is
just one multiplied by the inverse of that fraction. So since we had 1 over 5/4, that's
the same thing as 1 times 4/5, which of course is just 4/5. So we have 2 times 4/5, plus
2 times 0 squared, which is just 0 so we have 1/1 which is just 1. So of course, 2 times
1 is just 2 so I'm just gonna leave that as 2. We did that one, that one and that one.
1/2 squared is 1/4 plus 1 is 5/4, 1 divided by 5/4 is 4/5 so plus 2 times 4/5 again. 1
squared is 1. We've got 1 plus 1 is 2 so that's ½, so plus 2 times 1/2 is just 1. 3/2 squared
is going to be 9/4 plus 1 so we've got 4/4 plus 9/4 is 13/4, 1 divided by 13/4 is the
same as 4/13. So plus 2 times 4/13. And then 2 squared is 4 plus 1 is 5. 1/5 there and
of course that whole thing, 1/2 divided by 2 is just 1/4.
So I know that that simplification was quick but you guys can do the same thing on your
calculator. Trapezoidal problems, almost always end up using your calculator, you almost always
end up with a decimal because the fraction is really ugly. So you can just do the whole
thing on your calculator but I'm showing you guys on my hand. So 1/2 plus 8/5 plus 2 plus
8/5 plus 1 plus 8/13 plus 1/5 times 1/4. So in this case, I could simplify. I'll do
1/2 and then combine this 8/5, 8/5, and 1/5. So that's 17/5. I'm just gonna do this because
it's easier to plug in to my calculator and then I have 2 plus 1 is 3 plus 8/13 times
1/4. So now that I've gotten this down to something
that's easier, I'll go ahead and plug this into my calculator. So I'll just do 1 divided
by 2 plus 17 divided by 5 plus 3 plus 8/13 and I'll get that answer and I will just divide
that whole thing by 4, just the same thing as multiplying by 1/4. So I divide that whole
thing by 4 and I get as my final answer, 1.8788. And it gives you more decimal places but you
could round to four. I think that would be sufficient. So I know it may be a little bit
confusing; a lot of separate on the board but that is basically in a nutshell how you
apply the trapezoidal formula and you're final answer is 1.8788. Thanks, guys. See you next
time.