Hi everyone.
Welcome back to integralcalc.com.
We're going to be another trapezoidal rule problem today.
This one, the function that we're given is the integral from 0 to 1 of 1 over e to the
x plus 1 dx where n equals 10.
And we're asked to use the trapezoidal rule to approximate this function.
So I wrote down the formula for the trapezoidal rule in the last video but I think it's just
so ridiculous that I find it unnecessary to actually write it down.
I think it cause more confusion than anything.
So I’m not going to actually write down the whole formula.
I’m just going to go through how to solve the problem.
The first thing we need to do is find delta x, which is really simple to do.
The way that we find delta x is by subtracting, we do the top number here, 1 and we subtract
the bottom number here on the range, 0.
And we divide by n.
So we get 1 minus 0 which is 1 over 10 which is the same thing as 0.1.
So that's the first thing we do.
We find delta x.
We find the top number minus the bottom number, divide it by n.
And basically what that tells us, n is the number of trapezoids that they want us to
use to approximate the area under this graph or under this curve.
So we're going to break it up into 10 trapezoids.
And in order to do that, we need to divide the area from 0 to 1 into 10 sections.
That means when we find delta x, the first trapezoid is going to go from 0 to point 1,
the second trapezoid is going to go from point 1 to point 2, the third one from point 2 to
point 3, all the way up until the last one which goes from point 9 to 1.
Hope that makes sense.
So once we find delta x (which is really simple to o), the next thing that we want to go ahead
and do is write down the areas where the trapezoids begin and end.
So we write down x0.
x0 is always the bottom number here or the left-hand side of the range, so x0 is 0, x1,
then we start adding delta x each time.
So 0 plus point 1 is 0.1 and we just keep adding delta x until we get to the second
number or the top number in the range.
So we're going to add point 1 each time.
x3 equals 0.3. x4 equals 0.4. x5 equals 0.5. x6 equals 0.6. x7 equals 0.7. x8 equals 0.8.
x9 equals 0.9 and x10 equals 1.
And now that I have reached the number of the range, the right-hand limit of the range
or just the top number here on this integral, I can stop.
I’ve got all of my values of x.
So first step find values of x, second step write out all these things.
First number should be the bottom number here.
Last number should be the top number here.
And everything else in between, you just add delta x each time until you get here.
So we've done that.
Third step is to apply the trapezoidal rule.
And basically what that looks like (we'll write that down), you just plug in each of
these to our function here.
So it's going to be super long but we'll go ahead and write it down.
So first that we're plugging in is 0 into this function here.
So we just do 1 over e to the 0 plus 1.
See how we plugged in 0 here for x?
Now we're going to plug in point 1, so plus 1over e to the 0.1 plus 1 and we can just
go ahead and keep doing this all the way to the bottom.
So this is going to be tedious.
0.2 plus 1, plus 1 over e to the 0.3 plus 1, plus 1 over e to the 0.4 plus 1 (and this
is where I'm going to stop talking because you're probably getting annoyed with my voice
and I think you get the point).
And finally you can see we've arrived at 0.9, we're going to do 1 over e to the 1 plus 1.
It's the last term.
Since we've hit the last term, we're done.
We can stop we close it with the bracket.
So you plug in all these values to this function here and you just add them all together.
Now, once you've done that (I left myself a little bit of space), the trapezoidal rule
says that you multiply every term in here except for the first one and the last one
by 2, so I'll go ahead and put 2, 2 because we're going to multiply each one of these
by 2 but not the first one and not the last one.
So you can write everything out and then go back in and add 2 to all of that.
So that's most of the formula except the last thing you have to do, on the outside of these,
we multiply by delta x which we got was 0.1 over 2.
The 2 always stays the same.
It’s just delta x over 2.
You multiply that on the outside and that is the entire formula.
So first step, find delta x.
Second step, write out all of these x terms here.
Third step, plug them all into your function and just add them all together, multiply each
term inside the brackets except the first one and the last one by 2 and then delta x
over 2 on the outside.
And now all we need to do is simplify.
And then use a calculator to solve.
So I'm going to try to simplify this as quickly as possible.
We have e to the 0 is 1.
Anything raised to the 0 power is 1.
So 1 plus 1 is 2, so the first term here is a 1/2.
then we've got (I’m just going to write this out), 2 over e to the 0.1 plus 1 plus
2 over e to the 0.2 plus 1 plus 2 over e to the 0.3 plus 1 (and that's where I'm going
to stop talking).
And then remember we don't multiply the first term or the last term by 2 so this is just
e to the 1 is e so this is just plus 1 over e plus 1 and then the whole thing, point 1
divided by 2 is point 05.
So I hope you guys can see all of that.
But that is pretty much as simplified as we're going to get.
And now, you would pick up your calculator and you would (let me see if I can write this
out for you so I make sure that it's right) plug this whole thing into your calculator.
So the beginning is going to look like this.
Let me see.
Okay, so in your calculator, this is going to look like, if you have like a TI '89 or
later than, you get to see more detail but if you've got an older calculator, which I
do, my problem in my calculator is starting to look like this.
1/2 +2/e­0.1 +1 and yeah, I’ve got this.
So I’ve got you know 1/2 plus 2 over and then the whole denominator here and then I’ll
have plus 2 over e to the point 2 plus 1 and then I’ll keep going.
And then because I realized that I was going to run into this problem; it's going to be
extremely tedious, I went ahead and did the calculation ahead of time so that you didn’t
have to waste your time.
The final answer that you should get when you do this whole calculation and then of
course go ahead and hit enter and then multiply the whole thing by point 05, when you finish
the whole thing, the answer you should get is
0.37993, approximately with some more decimal places but that's rounded so that will be
your final answer.
You know it's a lot of work on the calculator but you just can't make this simpler without
busting out the old calculator.
So go ahead and plug that all in.
Be really careful about your parenthesis so you make sure that you don't miss anything
or that the calculator does not misinterpret what you’re trying to plug in and see if
you get 0.37993.
Thanks guys.
I’ll see you next time.