Hi, everyone! Welcome back to integralcalc.com. Today, we’re going to be doing a problem
with Simpson’s approximation. And in this particular problem, we’ve been asked to
use Simpson’s approximation to find the area under the curve x^2 on the range 1 to
3. So you can see here we’ve been given the integral here and we know by looking inside
the integral that the function we’re concerned with is x^2. And because we have these limits
of integration, 1 and 3, we know that we’re going to be evaluating on the range 1 to 3.
They’ve also told us that we’re going to be dividing the range 1 to 3 into four
subintervals because n = 4. The one point to remember about Simpson’s approximation
is that you can only use it when you divide the range into an even number of subintervals.
So you can divide your range 1 to 3 into 2, 4, 6, 8, 10, whatever subintervals but you
couldn’t divide it into 1 or 3 or 5 or 7 subintervals. It must be an even number of
subintervals. So in this case, we’re dividing it into
four so we know that Simpson’s approximation will work for us. Now we need to find all
these values that are in the Simpson’s approximation formula so that we can just plug them into
the formula. So the first one we see here is delta x. and to find delta x, we’ve got
basically this range here a to b. Delta x is (b – a)/n so if we plug in 3 for b and
1 for a, we get 3 – 1 divided by, we already know that n = 4 so we plug in 4 and that gives
us obviously 2/4 which we can see is 1/2. So we’ll be plugging in 1/2 for delta x
into our Simpson’s approximation formula. So now, I think the easiest way for me to
remember what I need to plug into the Simpson’s approximation formula, I like to draw a number
line on my range. So we already know we’re going from 1 to 3. So I draw a number line,
I write my end points 1 to 3 and now that I’ve calculated delta x is 1/2, I go ahead
and write in my integrals. So since delta x is 1/2, we need to start with 1 and add
1/2 to it until we get up to 3. So 1 + 1/2 = 3/2, 3/2 +1/2 = 2, 2 +1/2 = 5/2 and 5/2
+ 1/2 = 3. So we get up to 3. So that’s how I break out my number line. So I just
started with my lower limit of integration 1 and added delta x to it each time until
I got to my upper limit of integration, 3. So now I’ve divided my number line. I’m
going to need each one of these values from my Simpson’s approximation formula so that’s
why I like to write them out. So I’ve got these values. Remember that
with Simpson’s approximation, we don’t multiply our end points 1 and 3 by anything
special. But you can see here in the formula, we’ve got y sub zero and we’ve got y sub
n. not multiplied by any coefficient but you can take one step in and you can see we’ve
multiplied by 4 and we’ve multiplied by 4. I’m moving from both the left and right
side, moving in toward the middle of the formula and we multiply by 4 for the second term in.
Then if we take one more step in, we can see we’re multiplying by 2 and then by 2. If
we take one more step in, by 4 and then by 4 and then by 2. So basically, if you start
from left to right, we’re multiplying by nothing then by 4 then by 2 then by 4 then
by 2 until we get to the end. The last thing we’ll multiply by 4 and then by nothing
again. So it’s kind of confusing but basically, when I’ve got my number line, I don’t
multiply 1 and 3 by any thing but I know I’m going to multiply 3/2 and 5/2 by 4 and then
I’m going to multiply 2 by 2. And if I had more sub intervals, I would start from the
outsides, multiplying by nothing, then move in one step on both sides and multiply by
4, move in both sides one step and multiply by 2, move in both sides one step and multiply
by 4, and keep going until I get to the center. So I go ahead and write that out from my formula
and then it’s just really a matter of plugging these in so we’re going to take all these
values and we’re going to evaluate them in our function. Our function is x squared.
We know from our original function is x squared. So basically, Simpson’s approximation S
of n and we can actually write this as S sub 4 because we know that n = 4. So Simpson’s
approximation of this function using four sub-intervals is going to be delta x which
we know to be 1/2 over 3. And then we’re going to multiply by y sub zero. This y sub
zero in our formula, we already know that it’s 1 and that we’re not multiplying
it by anything but we need to plug all these values into our function, x squared. So plugging
1 in to x squared, 1 squared would just give us 1. Then we add to that 4 times 3/2 in the
function so basically plus 4 times f of 3/2. Then we say plus 2 times f of 2 plus 4 times
f of 5/2 and then plus f of 3. When we plug 3 into our function, x squared, 3 squared
will just be 9 so we’ll get 9 there. And that’s it. Now we can just evaluate
this. So S sub 4 equals 1/2 divided by 3 is going to be 1/6 and then when we simplify
what’s inside here, 1 plus 4 times, when we plug 3/2 into x squared, we get 3/2 squared
which is just going to be 9/4. Plus 2 times f of 2. 2 squared is just 4 so we end with
just 2 times 4 plus 4 times 5/2 squared which is just going to give us 25/4 plus 9. And
now we’ll get 1/6 times 1. These 4s will cancel so we’ll get plus 9. And then 2 times
4 is 8 and then these 4s will cancel so we’ll get plus 25 plus 9. And when we simplify what’s
inside our parenthesis, 1 plus 9 is 10, plus 8 is 18, plus 25 is 43, plus 9 is 52 so times
52 which is going to be 52/6. We can divide those both by 2 and we get 26/3. And that’s
it. That’s our final answer using Simpson’s approximation.
So remember the easiest thing to do is write out your number line, put in your limits of
integration 1 and 3. Those are your end points. Once, you’ve got delta x, you can start
adding that to your lower limit of integration. So you add it to get to 3/2, add it to get
to 2, add it to get to 5/2, add it and you get to 3. So you realize you’re at the end
of the line. You’ve now got all of those points at which you are going to need to evaluate.
You’re then going to evaluate all of these points 1, 3/2, 2, 5/2 and 3 and then you’re
going to multiply it by whatever these factors are down here. So these ones obviously, you
just multiply by 1. It’s just like multiplying by nothing and then you move in one step and
multiply by 4, move in another step, multiply it by 2. You could keep going. But then you’re
going to take those and plug them into your formula. So obviously, we’ve plugged in
delta x here for 1/2. 3 is just part of the formula and then you’ve got this one multiplied
by 1 and multiplied by 1 multiplied by 4 and 4 and then multiplied by 2 each of our points.
And then it’s just a matter of using arithmetic to simplify and solve.
So Simpson’s approximation tells us that the area underneath our curve, and if we just
wanted to get a visual, we know that the curve is x squared so that’s going to look roughly
like this and we’ve got our range 1 to 3. So what we found was this area in here underneath,
between the curve and the y axis on the range 1 to 3. So Simpson’s approximation told
us that that area is roughly 26/3. So that’s it. I hope that video helped you guys and
I will see you in the next one. Bye!