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.