Numerical Integration With Trapezoidal and Simpson's Rule

Summary

This educational video script delves into numerical integration, a method for approximating definite integrals that are difficult to solve analytically. It introduces two primary techniques: the trapezoidal rule and Simpson's rule. The trapezoidal rule is explained through a step-by-step process, emphasizing the calculation of ΔX, the use of function evaluations at incremental points, and the summation formula. The script then contrasts this with Simpson's rule, which alternates between multiplying function evaluations by four and two, requiring an even number of intervals for accuracy. Practical examples, including calculating integrals of 1/x from 1 to 2 and 1/√(x+1) from 0 to 2, demonstrate the application of these methods, showing how they can closely approximate true integral values, even when exact solutions are unknown.

Takeaways

• 🔢 Numerical integration is a method used to approximate definite integrals that cannot be solved using traditional calculus techniques.
• 📐 The Trapezoidal Rule is one such method, which approximates the area under a curve by dividing it into trapezoids rather than rectangles.
• ∆x The Trapezoidal Rule formula involves calculating the sum of function values at intervals (∆x) multiplied by their respective coefficients and then taking the average.
• 📉 The Simpsons Rule is another numerical integration technique, which uses a more complex formula involving the sum of function values at intervals multiplied by coefficients that alternate between 4 and 2.
• 🔁 Both the Trapezoidal and Simpsons Rules require the number of intervals (N) to be specified, with the Simpsons Rule needing an even number of intervals for accuracy.
• 📋 The script provides a step-by-step example of how to apply the Trapezoidal Rule to approximate the integral of 1/x from 1 to 2 using 10 intervals.
• 📘 The script also demonstrates the application of the Simpsons Rule using the function 1/√(x+1) from 0 to 2 with 7 intervals, emphasizing the need for even intervals.
• 🔍 The accuracy of numerical integration methods improves as the number of intervals increases, leading to a better approximation of the actual integral.
• 📊 The script compares the results of the numerical integration methods with the actual integrals when possible, showcasing the closeness of the approximations.
• 💡 The script serves as a tutorial for students who may not have covered certain integrals in their calculus courses, providing an alternative approach to finding area under curves.

Q & A

What is numerical integration?

Numerical integration is a method used to approximate definite integrals that cannot be solved analytically, providing a way to estimate the area under a curve when an exact formula is not known or available.

What are the two main methods of numerical integration discussed in the script?

The two main methods of numerical integration discussed in the script are the trapezoidal rule and Simpson's rule.

How does the trapezoidal rule work?

The trapezoidal rule works by approximating the area under a curve by dividing the area into trapezoids rather than rectangles. It sums the areas of these trapezoids to approximate the definite integral.

What is the formula for the trapezoidal rule?

The formula for the trapezoidal rule is given by \(\Delta x \frac{f(x_0) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(x_n)}{2}\), where \(\Delta x\) is the width of each trapezoid, and \(f(x_i)\) represents the function evaluated at the points \(x_i\).

What is the significance of the term 'Delta X' in the context of numerical integration?

In numerical integration, 'Delta X' (\(\Delta x\)) represents the width of the subintervals into which the area under the curve is divided to approximate the integral using either the trapezoidal rule or Simpson's rule.

Why is the number of terms (N) important in numerical integration?

The number of terms (N) is important because it determines the accuracy of the approximation. A larger N results in more subintervals, which generally leads to a better approximation of the integral.

How does Simpson's rule differ from the trapezoidal rule?

Simpson's rule differs from the trapezoidal rule in that it approximates the area under a curve using parabolic segments instead of trapezoids. It uses a different weighting for the function evaluations, with the first and last terms having a coefficient of 1, the middle terms having a coefficient of 4, and every other term having a coefficient of 2.

What is the formula for Simpson's rule?

The formula for Simpson's rule is given by \(\frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + \ldots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)]\), where n must be an even number.

Why does Simpson's rule require an even number of terms (N)?

Simpson's rule requires an even number of terms because it relies on the parabolic shape that is formed between every two points, and this requires pairs of intervals to create the parabolic segments for the approximation.

How can one improve the accuracy of numerical integration using the trapezoidal or Simpson's rule?

The accuracy of numerical integration can be improved by increasing the number of terms (N), which results in smaller subintervals and a more detailed approximation of the curve's shape.

What is the practical application of numerical integration?

Numerical integration is used in various fields where exact integration formulas are not available or practical, such as in physics for calculating work done, in engineering for stress analysis, and in computer graphics for rendering.

Outlines

00:00

📚 Introduction to Numerical Integration

The speaker begins by introducing numerical integration as a method to approximate definite integrals that cannot be solved using traditional calculus techniques. They mention that there are integrals that are not covered in a standard calculus curriculum, and numerical integration provides a way to estimate their values. Two main methods are discussed: the trapezoidal rule and Simpson's rule. The speaker emphasizes that these methods are approximations and are useful for integrals that are otherwise difficult to compute.

05:00

📐 The Trapezoidal Rule Explained

The speaker delves into the trapezoidal rule, explaining it as a technique to approximate definite integrals by dividing the area under the curve into trapezoids instead of rectangles. The formula for the trapezoidal rule is introduced, which involves summing the function values at intervals and multiplying by a factor that includes the change in x (Δx). The speaker clarifies the concept of Δx and how it's calculated as the interval width divided by the number of intervals (N). An example is given to demonstrate the calculation process, where the integral of 1/x from 1 to 2 is approximated using N=10, resulting in 11 terms to be calculated.

10:03

🔢 Applying the Trapezoidal Rule with an Example

The speaker applies the trapezoidal rule to the example of integrating 1/x from 1 to 2 with N=10. They calculate Δx as 0.1 and then determine the x values for each term in the sum, incrementing by Δx for each subsequent term. The formula is then applied by multiplying the sum of the function values at these x points by the appropriate coefficients and Δx/2. The speaker emphasizes the importance of following the formula correctly, especially with the coefficients of 2 for all terms except the first and last.

15:03

📉 Simpson's Rule: An Alternative Method

The speaker contrasts Simpson's rule with the trapezoidal rule, highlighting that Simpson's rule provides a more accurate approximation by using a different set of coefficients. Simpson's rule divides the interval into smaller pieces and uses a combination of 4s and 2s as coefficients, but it requires an even number of intervals. The speaker provides a new example using the function 1/√(x+1) from 0 to 2 with N=6, illustrating how the x values and the coefficients are determined and applied in the formula.

20:05

🔍 Comparing Numerical Integration Methods

The speaker compares the results of the numerical integration methods with the actual integral values when possible. They show that the trapezoidal and Simpson's rules provide close approximations to the true integral values, with the accuracy improving as the number of intervals increases. The speaker concludes by emphasizing the utility of these numerical methods for approximating integrals that cannot be solved analytically.

25:56

🎓 Conclusion and Understanding Numerical Integration

In the final paragraph, the speaker wraps up the discussion on numerical integration, ensuring that the audience understands the concepts of the trapezoidal and Simpson's rules. They encourage the audience to practice these methods and to check their work against known integrals to gauge the accuracy of the approximations. The speaker's tone is light-hearted, indicating that the session was meant to be educational and interactive.

Keywords

💡Numerical Integration

Numerical integration is a method used to approximate the value of a definite integral when an analytical solution is either too complex or impossible to obtain. In the context of the video, numerical integration is central to the theme as it's used to demonstrate how to approximate integrals that cannot be solved using traditional calculus techniques. The video discusses two specific methods for numerical integration: the trapezoidal rule and Simpson's rule.

💡Trapezoidal Rule

The trapezoidal rule is a numerical integration technique that approximates the area under a curve by dividing the area into trapezoidal sections rather than rectangular ones. The video explains this concept by stating that the rule involves summing the function values at intervals and multiplying by the width of the interval divided by 2. It is used as a method to approximate integrals when exact solutions are not known, as illustrated by the example of approximating the integral of 1/x from 1 to 2 with N equals 10.

💡Simpson's Rule

Simpson's rule is another numerical integration technique that provides a more accurate approximation than the trapezoidal rule by using parabolic sections instead of trapezoidal ones. The video highlights this method as an alternative to the trapezoidal rule, mentioning that it involves multiplying the function values by different coefficients (four and two) and summing them, using the formula Delta X divided by 3. It is noted that Simpson's rule requires an even number of intervals.

💡Delta X

Delta X (∆x) is a term used in numerical integration to represent the width of each subinterval when the area under a curve is divided into smaller sections for approximation. In the video, Delta X is calculated as (b - a) / N, where 'b' and 'a' are the bounds of integration and 'N' is the number of subintervals. It is a crucial component in both the trapezoidal rule and Simpson's rule, as it determines the granularity of the approximation.

💡Definite Integral

A definite integral is an integral that has both an upper and lower limit, representing the signed area under a curve between two points. The video script discusses the concept of definite integrals as the quantities that numerical integration methods, such as the trapezoidal rule and Simpson's rule, are used to approximate. The definite integral is integral to the theme of the video as it sets the stage for the need of numerical approximation techniques.

💡Approximation

Approximation in the context of the video refers to the process of estimating the value of a mathematical quantity, such as a definite integral, when an exact value cannot be easily computed. The video emphasizes the use of numerical integration techniques as a means of approximation, particularly when dealing with complex or unknown integrals. The trapezoidal rule and Simpson's rule are both presented as methods for approximating definite integrals.

💡Integrals

Integrals are a fundamental concept in calculus that represent the area under a curve and are used to solve a variety of mathematical and real-world problems. The video script discusses integrals in the context of numerical integration, explaining how certain integrals cannot be solved using standard calculus techniques, thus necessitating the use of numerical methods like the trapezoidal rule and Simpson's rule.

💡Calculus 1

Calculus 1 is typically the first course in a sequence of calculus classes and covers foundational concepts such as limits, derivatives, and integrals. The video script references 'Calculus 1 days' to set a context for the level of mathematical knowledge the audience might have, implying that the audience has a basic understanding of calculus but may not have encountered more advanced techniques like numerical integration.

💡Late Transcendentals

Late transcendentals is a term used in some calculus courses to refer to the study of certain integrals that are covered later in the curriculum, often in a second semester or a more advanced class. The video script mentions 'late transcendentals' to explain why certain integrals, like the one involving 1/x, would not have been covered in a first-semester calculus course, thus justifying the use of numerical integration for approximation.

💡Iterations

Iterations in the context of numerical integration refer to the individual steps or calculations performed within the process of approximating an integral. The video discusses the concept of increasing the number of iterations (increasing 'N') to improve the accuracy of the approximation. More iterations mean more subintervals and thus a finer approximation, as demonstrated by the comparison of results with different values of 'N'.

Highlights

Introduction to numerical integration as a method to approximate definite integrals that are difficult to calculate directly.

Explanation of the trapezoidal rule as the first method for numerical integration.

Description of the trapezoidal rule formula and its components, including Delta X and the sum of function values.

Clarification that Delta X is calculated as (B - A) / N, where B and A are the bounds of integration and N is the number of intervals.

Emphasis on the importance of N being a finite number for the trapezoidal rule to provide an approximation.

Introduction to the concept of X sub i, representing the points at which the function is evaluated.

Example calculation using the trapezoidal rule to approximate the integral of 1/x from 1 to 2 with N equals 10.

Step-by-step guide on how to calculate each X sub i and the corresponding function values.

Discussion on how increasing the number of terms (N) improves the accuracy of the trapezoidal rule approximation.

Introduction to Simpson's rule as an alternative method for numerical integration.

Explanation of the differences between Simpson's rule and the trapezoidal rule, including the use of Delta X / 3 and the alternating coefficients.

Requirement for N to be an even number when using Simpson's rule.

Example calculation using Simpson's rule to approximate the integral of 1/sqrt(x + 1) from 0 to 2 with N equals 6.

Comparison of the calculated approximations with the actual integral values to demonstrate the accuracy of numerical integration methods.

Conclusion on the effectiveness of numerical integration methods like the trapezoidal rule and Simpson's rule for approximating difficult integrals.

Transcripts

okay for us our very last section

talking about numerical integration kind

of going back in time to our calculus 1

days or first semester calculus here's

what numerical integration is all about

what it allows you to do is approximate

integrals definite integrals that you

couldn't otherwise do that we don't have

a formula for that man we I don't know

how to do this one or something we

haven't covered because there are going

to be some integrals that we have not

covered even though we've done all our

techniques integration there's somebody

looking up I don't know as a possible

action no it's a trick I don't know well

we can approximate those definite

integrals with these two processes one's

called the trapezoidal rule and one code

is called Simpsons rule so we're going

to talk about trapezoidal rule first

then Simpsons rule they're very similar

and what you do what this is doing again

it's giving you an approximation of a

definite integral without having to even

do the integral you with me okay so like

I said I think I just mentioned this to

you guys but put yourself back into your

first semester calculus days when we

haven't done a lot of integrals okay so

the ones we're going to do on the board

they would be something you learn in

calculus two what we've all we've

already learned them but back then these

integrals would have not been really

possible for us because we would not

have learned them yet you with me okay

so with that in mind let me introduce

you the trapezoidal rule

so the trapezoidal rule says this if you

want to approximate a definite integral

remember the definite integrals have

bounds of integration they go from like

an e to e they go somewhere of any

function that you want we can

approximate it hits our squiggly equals

here we could we can approximate it by

doing this and really interesting here's

how it works

it says I don't define these in just a

minute but they should look familiar if

you took calculus 1 this looks familiar

Delta X you guys have seen Delta X

before Epogen will talk without the

finance second if you don't remember

what it is you take Delta X u divided by

2 and you multiply it by this sum by f

of X sub 0 plus 2 times f of X sub 1

plus 2 times f of X sub 2 plus and you

keep all those twos and tell you get

down to 2 times f of X sub n minus 1

plus f of X sub n no - that's the

trapezoidal rule now I'm going to define

a couple of these things for you

personally what Delta X is if you don't

remember it's okay Delta X is B minus a

where do I get my B minus a from do you

know that's a certain iman say it's just

your your integral so B minus a divided

by N and it's going to be given to you

so n is going to be some finite number

because we're approximate right now now

listen we can't let n go to infinity

because if we did well we wouldn't have

an approximation anymore and this is

going to we would have a finite number

of terms add up and one via

approximation we can't actually do that

we couldn't do this forever the idea of

doing this forever is the actual

integrals

add so because do the integral well then

we're left with a finite number of terms

to approximate our value and then X sub

I which means that X sub 0 X sub 1 X sub

2 X sub 3 X + 4 X sub n minus 1 X sub n

it's given by this

you start with Eddie again a is just

right here and you add to it your index

times Delta X that might look a little

confusing you get a treasure on this

this is not that hard to do

I'm going to prove that to you right now

with an example so write down the

formula I'm going to show you how to do

it just make sure you write down

correctly okay there's no two in front

of this there's a 2 in front of every

other term that you have until you get

to the very very last one and then

there's not a 2 there quick hit up

you're ok with that so far okay so let's

do our example so what we're going to do

is we're going to approximate 1/2 - 1

over X DX and what we want to do is

approximate this with N equals 10 if N

equals 10 what that's going to mean for

us is that we're going to get 11 terms

here we have an X sub 0 through X sub 10

we're going to end on the X sub 10 term

does that make sense to you so we need

eleven terms stop it N equals 10 now if

you look at that many of you're going to

think well I'll just do the integral

what's the integral of 1 over X would

you know that in a late transcendentals

class like we have for calculus 1 as I

kept this one we don't cover that so

that's why I said put yourself back into

calculus want first semester calculus

class at our College we have what's

called late transcendental some people

have early where they do this in

calculus 1 we don't so we do it later so

for us this would have been impossible

back in calculus 1 so all we would have

done is try to approximated here's how

we're approximated the first thing we do

we'll figure out Delta X of course

that's going to be important for us so

to figure out Delta X Delta X is B minus

a over N all those things are going to

be given to you if you have a definite

in a roll what's our beat - over and

perfect that's going to be 110 okay what

we're close to place and putting this in

our formula now we got to figure out our

X sub 0 or X sub 1 X sub 2 - X sub 3 and

X sub 4 blah blah blah in fact you can

think of this is because I'm going to be

some calculators you can think about

this is 0.1 if you want to you might

make things a little bit easier for you

if you want so it looks kind of weird

like how am I going to figure it's not

that hard here's what it says if I want

X sub 0 X sub 0 says start at a would

say add 0 D X's so we're just going to

have one does that make sense

exit sorry X 1 1 missed one exit 1 says

this started a what's what's a 1 and add

1 Delta at one time is Delta X so it'd

be one plus point 1 or just 1 point 1 X

sub 2 says start a it's 1 plus 2 times

Delta X well that would be 1 plus 0.2

1.2 can you tell me what the next one's

going to be

going through what's the next one and

then 1.5 and 1.6 then 1.7 1.8 and 1.9 at

the very end we're going to get to X sub

9 that's going to be one point and then

X sub 10 gives you I wish so hands feel

okay with where those are coming from

can you follow that should be a little

easier because it's back to factor this

one stuff it just plug in numbers in it

just says hey start here start whatever

that is and just add Delta X so for X

sub 0 yet none of them so just one and

then add this one point one add it again

one point two add it again

1.3 1.4 1.5 1.6 1.7 1.8 1.9 and then to

one point n - yeah so we get to now what

the trapezoidal rule says is after

you've all if you've done all that the

integral can be approximated by doing

this you take Delta X what's our Delta X

so we have Delta x over 2

and then you're going to do this you're

going to have f of X sub 0 F of what

plus what's the next thing I'm going to

right now our to store two times F of

what yeah plus 2 times 1 point 2 plus 2

times F of 1 point 3 plus the reviews

all the way down till we get to 2 times

F of 1 point 9 and then we're going to

have do we have a two on the very last

term no oh just F of 2

okay I want to do like a 20-second recap

just to make sure that you give this so

the idea behind these numerical

integrations is sometimes we can't

integrate or sometimes we don't know how

to integrate it and this would be one of

those cases where at the time we would

not have known how to do this now I'm

giving you a very simple example okay

just so it makes our math easy so you

get the idea you could do this with some

pretty hardcore integration I know that

you don't know how to integrate if it's

a definite integral to understand that

if it's not a definite integral this

this doesn't work because I'm plugging

numbers in alright so we say is all

right now where's your integral starting

whatever where it is stopping whatever

you can find Delta X with that provided

you're given the number of iterations

and your your your rule so in this case

we have N equals 10 that's 11 terms that

we're going to be going through so it

says okay cool well if that's my Delta X

then I start with my a I just add my

Delta X for every new term so I added

them add it than that until I get up in

my last one plug in the formula we have

Delta x over 2 no problem f of X of 0 f

of X sub 1 so 1 1 will point you all the

way to the very end notice though into

two terms that don't have two s on the

first one and the last one you follow

now what's nice about this do you have a

function that you can plug these things

into yeah that's right there it's just 1

over X so we're going to do this we're

going to start by taking our Delta X

it's 0.1 divided by 2 and then we're

going to have F of 1 so that's 1 over 1

plus 2 times F of 2 so check it out it's

going to be 2 times 1 over 1 point 1

plus two times one over one point two

plus two times one over one point three

plus we do this for a while for a finite

number of terms and we're going to end

off with two times one over one point

nine plus one over two I want you to see

if you can you can go through that you

mine data that actually makes sense to

you so fans if that does make sense to

you okay so here's what you're going to

do on your calculators what you're going

to do is you're going to figure out what

all these things are I'm not writing all

because it takes too long so but I'm

hoping that I've made this clear on what

this this is you guys see what what it

is we're starting with just a function

evaluated at whatever RNA is just one

and then we add to 2 times F at one

point 1/2 times F at 1 point 2 2 times F

of whatever our exit 3 exit for X upon

all the way up till we do our X sub n

minus 1 look at that X is 1 is 9 some of

you guys on your tested X sub n plus 1

for this one you were counting further

than you should have been counting

before not after

so X sub n minus 1 gives us our 9 and

then our last one whatever it is in 10

in this case no to here we plug them

into our function that's what we have

our one over this one over this one over

X of 3 1 over X sub 10 and we just

figured decimal equivalent so in our

case on your calculator you get 1 plus 2

over this it's 2 into this through that

and then multiply that by 120

now as you're doing that I'm going to go

ahead on actually I did this on purpose

I did the integral because we can

actually do it I want to see how close

this actually is so if we were to really

do the integral so people work on this

some of you guys really go through this

okay I want I want an actual answer so

but if we were to do an integral with

what we know now we know the integral of

1 or X is Ln X and we would evaluate

that from 1 to 2 that's Ln of 2 minus Ln

of 1 well ml element 1 is 0 so we get Ln

of 2 + Ln 2 is about equal to someone

who's not working on this can you tell

me what Ln of 2 is about equal 2.6 died

and 3 14.6 not saving in 6 9 31 or okay

it keeps on going because it's it it

consider a rational number it goes on

forever without ever meeting have you

got this one what you get

yeah is it exactly the same is it close

tenths hundredths step it's the same to

the thousands it's off by a little over

six ten thousandths so with if you but

here's the point

this was clicker because we know how to

do it right if this was an integral that

we did know how to do this would be the

best way because you don't know how to

do it so you okay we'll call let's

evaluate this approximately let's use

our trapezoidal rule and then we get

something that's relatively close now

question how do you make this better

more terms na plus C cos is definite

integral how do you make it better

we're in Morgan's do that to like a

hundred it's really close I don't mean

very much but really close because

what's happening is you're taking our

interval and make it really really

living bill it's the same idea with

doing Vermont sounds with more and with

N equals larger numbers you're making

smaller rectangles smaller rectangles

well hey if you have smaller rectangles

that means you're missing less and less

area when you add up those rectangles if

you remember back to your your month

sums that you did in calculus or that

making sense to you so the major that is

the more close this is going to be I

just want to do this one to show you hey

you know what you can actually

proximately get it pretty darn close

dividends if I explain that long for you

okay this is it for trapezoid rule

that's how it works I'm going to give

you a super sense rule

show you how Simpsons rule differs

here's how Simpsons world works Simpsons

rule is going to look really really

similar here's only difference instead

of taking Delta X divided by two

you take Delta X you're going to divide

by three then what you do yeah it's cool

it's awesome electors - then what you do

is you take and you start with the same

f of X of zero but this becomes a four

that stays in tune and you're going to

go for two four two four two four two

four two and end of the pole right there

and then that last one doesn't get

anything now this will only work if n is

even and it's got to be even so you

can't do this if n is five you could do

it if n is six sort of ends eight or two

or like what even numbers are you do it

evens but other than that this small

report make sure I wrote down right to

it in from memory but yeah I think I got

it

years when do an example your real quick

yes because this makes sense to you that

we're going to alternate fours and twos

just you don't end the to you that

that's going to happen if you have an

eating okay so it's going to be nothing

then four two four two four two four two

four nothing thank you for your very

less terms will happen as me so let me

change this just a little bit

how about 1 over square root X plus 1

from 0 to 2 with N equals 7 is that ok

why not it has to be even find sticklers

is 6 and 6 go through the same way that

you would normally do the trapezoidal

rule only now we're just going to plug

into a different formula the Delta X is

calculated the same way the X sub i's

are calculated the same way and it just

has to be satisfying the tens even ok so

Delta egg what's nothing X is going to

be going to be 1/3 that's right I'd

probably leave that as a fraction 1/3

because I don't want I do not do not

want you to approximate in just 2.3 or

0.33 because you're going to be losing

something there ok and you're

essentially going to be losing it every

time you multiply every time you have a

term because of that distribution

property the approximation be way off so

leave this as a fraction unless you can

have a finite decimal you listening

right now okay so leave it as a fraction

okay well cool now let's do our X's so

we want X sub 0 we want X sub 1

like sub 2 and so on what's X sub 0

going to be what's X sub 0 say it louder

what you say 0 yeah you start with a 0 0

no problem ok well now if I have X sub 0

0 what's X sub 1 you know that one

it's give me 1/3 because you guys see

that the point in doing this well maybe

you don't see the point in actually

doing this we're trying to approximate

intervals integrals without actually

doing them we start with that number you

just keep on adding this your Delta X

per term that's what this says is that

start with your a Judas NRA is 0 right

now

you get it you start with a just add

Delta X that's it just add it every

single time

so what 0 plus 1/3 if we do know how to

do fractions so what's 1/3 plus 1/3 2/3

now 100 X sub 3 what's up 2/3 plus 1/3

3/3 1

what's X sub 4 4/3

what's X sub 5 5 thirds and what's X sub

6 6 thirds yay so it makes me locate

with that one so what I don't have all

the all the terms in this particular one

just so we feel through it again

sometime so our integral can be

approximated by doing this we start with

Delta X how much is done X 1/3 1/3 we

divided by what people carry very good

and then we have

this wholesome we have how's it start we

see the apps f of zero very good

f of zero then we add oh what's the next

thing for board starts before four times

f of 117 plus what 2 2 goes you know it

works

starts with nothing then it goes for

then it goes to F of 2/3

then it goes to what for fo and then it

goes back to 2 with a head for 4/3

then it goes back to for F of 5/3 and

then it is what please F of yeah but

notice there's nothing here so see how

it's going to work it's going to always

start with a full clips are done they

there with nothing start with 4 and with

a 4 and we're going to have one extra

force it's going to pair up

Bam Bam with an extra 4 it's not going

to go back to the deciding to get there

we use our even ends you're going to be

located with this one are you sure ok

and you can see if I went to N equals 8

what's going to happen is I'm going to

have another tuna to the four go

into tentative another children before

it's always going to end with that 4 and

then then that last term without

anything in front of it so what do we do

with all these numbers what are we to do

with them look you pull you into what

function yep right there so this is

going to be approximately 1 9 3 okay

1 over square root 0 plus 1 plus 4 over

square root of 1 3 plus 1 plus 2 over

square root of 2/3 plus 1 plus 4 over

square root of 1 plus 1 plus 2 over

square root of 4 thirds plus 1 plus 4

over square root 5 30 plus 5 plus

nothing 1 well not nothing 1 over square

root of 2 plus 4 and if you do that and

add them all up and then multiply by 1/9

then we get an approximation what I'd

like to know is a show fans if you

understand where all these things are

coming from 2 events if you do guys on

the right are you guys ok with this

ok so this is what I had to bring your

calculate in your calculator because

there's a what I can be able to do this

in our heads very easily so we're

approximate here where it's okay to get

our decimals so we got one that you add

that X square divide 4 by it and so on

and so forth and so on and so forth and

so on and so forth now I've already done

it you guys can go ahead and see if you

can check my work but just for the sake

of time it's going to take a little

while right so here's what it is what it

ends up giving you is approximately one

point four six four to one is what that

is

Japan's relocated with the idea now what

I want to do I want you guys to do the

interval because we do know how to do

this one we were kind of pretended we

did don't want you to do that integral

over here and just see what how good

that approximation is all right so go

forward

like you to check my work to make sure

that we all got the same thing did you

get about that I should just get exactly

that shouldn't know you got the two

square root of x plus one if you plug

back in the X before you start

evaluating perfect that's fantastic so

we got two or three minus two how much

is two times the square root of three

minus two one point eight four six four

101 is that one more time one point four

six four one zero one six okay that's

good enough that's our our decimal place

is it exactly the same is it close it's

really stinking close look at that this

is a one point four six four one this is

one point four six four two it's off by

a little over one ten-thousandth after

only six n equals six so six iterations

that's that's not too bad so that's

pretty darn close for approximately

something if we didn't know how to do it

so that's a lot of the trapezoidal rule

works that's how Simpsons rule works

have I explained it well enough for you

guys to understand yeah

oh wait no I'm just kidding