Trapezoidal Rule: Basic Form
Summary
TLDRThe script discusses the importance of integration in calculus, starting with the concept of areas under curves. It highlights Riemann's sum and the close relationship between integration and differentiation. The instructor explains why we need alternative methods like the trapezoidal rule and Simpson's rule for functions that are difficult to integrate directly. The trapezoidal rule is introduced as a way to approximate areas under curves by dividing the area into trapezoids, emphasizing its dependency on the function's nature for accuracy.
Takeaways
- đ The main topic discussed is integration, specifically the calculation of areas under curves, which is foundational to calculus.
- đšâđ Riemann's method of using sums and limits is highlighted as a clever and efficient approach to integration, closely related to differentiation.
- đ The script introduces the need for alternative methods of integration when dealing with functions that are difficult to integrate directly or when the function's rule is unknown.
- đ The trapezoidal rule is introduced as a method for approximating areas under curves, especially useful when exact integration is challenging.
- đ The trapezoidal rule is likened to the Riemann sum, but using trapezoids instead of rectangles to improve the approximation of areas.
- đ The formula for the area of a trapezoid is given as the average of the two parallel sides multiplied by the height, which in the context of integration is the integral from a to b of f(x)dx.
- đą The importance of understanding function values, specifically f(a) and f(b), is emphasized for calculating the heights of the trapezoids in the approximation.
- đ The script explains that the accuracy of the trapezoidal rule is highly dependent on the nature of the function being integrated.
- đŒ To improve the approximation, the script suggests using more trapezoids, effectively increasing the number of function evaluations and reducing the error.
- đ The lecture also mentions Simpson's rule as another method for integration that will be covered, indicating a progression from simpler to more complex techniques.
Q & A
What is the main topic being discussed in the script?
-The main topic being discussed is integration, specifically focusing on methods to calculate areas under curves, and the introduction of the trapezoidal rule as an approximation technique.
Why is the Riemann sum and the concept of limits important in the context of this discussion?
-The Riemann sum and the concept of limits are important because they form the foundation of the integration process, which is central to calculating areas under curves. They provide a method to approximate these areas using sums, which is essential when dealing with functions that are difficult to integrate directly.
What is the trapezoidal rule and how does it relate to the Riemann sum?
-The trapezoidal rule is a method for approximating the definite integral of a function by dividing the area under the curve into trapezoids instead of rectangles. It relates to the Riemann sum in that it's an extension of the idea of approximating areas by using simpler shapes, but it improves upon the Riemann sum by using trapezoids which can provide a better approximation.
Why might one need to use the trapezoidal rule instead of direct integration?
-One might need to use the trapezoidal rule instead of direct integration when dealing with functions that are difficult to integrate directly or when the exact form of the function is unknown, but measurements of the curve are available.
What is the formula for calculating the area of a single trapezium as described in the script?
-The formula for calculating the area of a single trapezium is given by \( \frac{1}{2} \times (f(a) + f(b)) \times (b - a) \), where \( f(a) \) and \( f(b) \) are the function values at points \( a \) and \( b \), and \( b - a \) is the width of the trapezium.
How does the number of trapeziums used affect the accuracy of the approximation?
-The more trapeziums used, the closer the approximation becomes to the actual area under the curve. This is because increasing the number of trapeziums reduces the error in the approximation, making the overall estimate more accurate.
What is the significance of the term 'function values' in the context of the trapezoidal rule?
-In the context of the trapezoidal rule, 'function values' refer to the values of the function at specific points, which are used as the heights of the trapeziums. These values are crucial for calculating the area of each trapezium and thus the total area under the curve.
Why is it important to know the function values when using the trapezoidal rule?
-Knowing the function values is important because they determine the heights of the trapeziums used in the approximation. Without these values, one cannot construct the trapeziums or calculate their areas accurately.
What is the potential drawback of using the trapezoidal rule for certain functions?
-The potential drawback of using the trapezoidal rule is that it can provide a poor approximation for functions with rapidly changing slopes or curves that are not well-represented by trapezoidal shapes, leading to significant errors in the calculated area.
What is the advantage of using the trapezoidal rule over other integration methods in certain situations?
-The advantage of using the trapezoidal rule is its simplicity and applicability when the exact form of the function is unknown or difficult to integrate. It allows for the estimation of areas under curves using only the function values at specific points.
Outlines
đ Introduction to Integration and the Trapezoidal Rule
The script begins by discussing the concept of integration, specifically the calculation of areas under curves. It mentions the historical development by Riemann, who used the idea of sums and limits to create calculus. The speaker emphasizes the precision and speed of this method, questioning why one would revert to less accurate and slower methods. Two reasons are given for learning alternative methods: first, the existence of functions that are difficult to integrate, and second, situations where the function itself is unknown, only measurements are available. The speaker introduces the trapezoidal rule as a method for approximating areas under curves, explaining that it is an improvement over the Riemann sum using rectangles. The process involves creating a trapezium by taking the tops of two function values and calculating its area using the average of the bases and the height, which corresponds to the horizontal distance between the function values.
đ Understanding the Trapezoidal Rule in Detail
This paragraph delves deeper into the trapezoidal rule, explaining how to calculate the area of a trapezium using the average of the two parallel sides (function values at points a and b) and the perpendicular height (the horizontal distance between a and b). The speaker clarifies terminology, such as 'function values,' which are the outputs of the function at specific points. The paragraph highlights that the trapezoidal rule's effectiveness depends on the type of function being analyzed. The speaker suggests that using more trapeziums can improve the approximation's accuracy, reducing the error. The process is generalized for multiple trapeziums, indicating that increasing the number of trapeziums results in a closer approximation to the actual area under the curve.
Mindmap
Keywords
đĄIntegration
đĄRiemann Sum
đĄDifferentiation
đĄTrapezoidal Rule
đĄFunction Values
đĄApproximation
đĄPrimitive
đĄArea Under a Curve
đĄNumerical Methods
đĄIntegration by Parts
đĄSimpson's Rule
Highlights
Integration is fundamentally about calculating areas under curves.
Riemann's sum and limits concept was pivotal in developing integration.
Integration is closely related to differentiation, offering a precise and efficient method.
The method's efficiency is why it remains the cornerstone of calculus.
The need for alternative methods arises when dealing with functions that are difficult to integrate.
Alternative integration methods become necessary when the function's rule is unknown.
The trapezoidal rule is introduced as a method for approximating areas under curves.
The trapezoidal rule is an improvement over the Riemann sum using rectangles.
A single trapezium can be used to approximate an area under a curve.
The area of a trapezium is calculated using the average of the bases and the height.
Function values at the endpoints are crucial for determining the heights of the trapezium.
The trapezoidal rule provides an approximation that can be improved by increasing the number of trapeziums.
The accuracy of the trapezoidal rule depends on the nature of the function being integrated.
Simpson's rule is mentioned as another method for approximating areas under curves.
The concept of function values is emphasized for its importance in calculating areas.
The trapezoidal rule is generalized for use with multiple trapeziums to improve approximation accuracy.
Transcripts
um
you will remember
this topic i've been calling it
integration right but really what it is
is um is areas under curves that's where
we started remember
and um
it just so happened that
riemann and his clever idea of riemann
sum and limits and calculus blah blah
blah blah blah worked out integration
this process was so close to
differentiation
was a really great way of doing it
it's not just great it's fast it's
precise you know um and that's why
because it's so good that's why we keep
using it like you know if you've got a
fast precise method it makes sense to
build your topic around that method okay
so being that we have a fast precise
method for calculating areas under
curves why would we go backwards
why would we come up with a method or
actually methods too um that are not
only less accurate
but as you'll discover most of the time
it's usually slower because
that is on integrated fireball okay now
two reasons number one um
we've so far been working with you know
functions where we know how to integrate
so
uh stuff like this
you know we can eat for breakfast yeah
right it's not that hard to think of
functions which we can differentiate but
we can't
integrate and you know it's not
complicated with example
suppose i wanted an area under this verb
now you've known this about this
function for months now it's not hard to
differentiate
how do you integrate and therefore find
the area underneath okay now at the
moment you guys can't
um
if you want to later on i'll teach you
how to integrate this function it
requires a fancy um a fancy method
really a fancier name called integration
by parts but the point is
like it's hard to work with okay but we
can still work out this area now firstly
what if my function is tough to
integrate secondly um what if i don't
even know what the function
is
like what if you've got
some measurements okay and what you want
is the area underneath okay but what
actually governs this
function you don't know what its rule is
okay you don't know what the x to the
power of this or whatever you've just
got measurements now think back when did
we learn how to deal with situations
like that yes questions you have
questions like this
and they're like here's a lake okay
we have we have these measurements
now work out what the area is okay so
yes
okay good question we'll get to i'm not
going to probably not send a cover today
but the short answer is
number one because you can it's not that
hard number two because it's useful like
you get blue shapes you can work out the
areas that's a handy skill
now anyway
this is why we're going to learn methods
for epoxy
we're going to learn two the first one
is called a trapezoidal rule
and um you might want to make that
subheading
the second one is called simpson's rule
but i anticipate we'll get to it
tomorrow actually i'm pretty sure
sorry
okay
so how does trapezoidal rule work um
suppose i want to find an area like this
so did we learn um
welcome to the present robbie okay now
if i want to find out what this area is
trapezoidal rule looks very similar to
the whole idea of the riemann sum which
is chopping up an area into you know
shape which i don't know intershapes
that do know okay now what did you use
for the testimony of right hand
rectangles right and when you turn it
into an integral then you just have a
whole button okay but it doesn't take
that much to realize that you can do
better than a rectangle
by using a trapezium okay so let's just
take the simplest case from a single
trapezium right if this is the area that
i'm after okay so
that's what i want
okay
i can make a trapezium that will be at
least in the right ballpark as this by
taking the tops of the two function
values okay and just drawing them
there's my trapezium
okay
now it's important to note that uh you
know these are vertical so that's a
right angle there right so how i go
about working uh calculating sorry the
area of this trapezium what is the area
of entropy
like if i had
the average of the bases
and then you take the average of the two
parallel sides okay now you might notice
i've deliberately not called them a and
b even though that's what we'd normally
call them why
yeah
because my a b and my lower bounds okay
so that's why i just
watch out so
n plus n
you add them and then taking on two
gives you the average and then you
multiply
so this is a standard way right now what
do each of these things correspond to in
our diagram where's the perpendicular
height of this trapezium
that horizontal distance in there okay
now in this case it's b minus a
it's a
which equals a
that's a more surprising result than i
thought it was okay so this b was a
right now what are these two parallel
sides equal to
well that's x equals a and that's x
equals b
and this is f of x okay what's the
height of the first one
and the second one is
now these are important um make a note
of this
we need to introduce in terminology the
quicker idea of the better because we'll
start to get confused later
um these guys f of a and f of b
right because i get these heights right
or you know bases if you like by
evaluating the function taking these
values and putting them into the
function okay we call them function
values
okay
the questions will refer to certain
numbers of function values so it's
important you know what they are
right now now that i've got my pieces
what's the area so i can say number one
the area is equal to the integral from a
to b
of
f
okay
but
i'm now going to sorry not equal to i'm
going to say that's approximately equal
to this trapezium that i've just
manufactured okay so it's this
perpendicular height on two so
b minus a whole one with two okay
and then i'm going to take uh the
average of the i don't know if it will
retain the image i'm just going to add
up the two
okay so you see normally we would have
said at this point you know what's
actually equal to if you know what the
primitive is
then you evaluate at your upper lower
bounds but the point is i don't know
what the primitive is or i can't be
bothered to calculate it so you can see
if we're going from here to here number
one is approximate number two i just
need to know what the original function
is this way
now as you can probably imagine um this
trapezium that we've just manufactured
might be a good approximation or it
might be a lousy approximation right so
if i give you a function like say
i don't know let's think of one um
i draw one down here something like this
okay and suppose i want to find its area
from say there's my a
um up to here to b right well the
trapezium you make out of this is like
five times the size of the area that i'm
actually after okay
trapezoidal rule um highly dependent on
you know what kind of function you are
as to how close you get right
but
i can still use trapeziums in a
situation like this to get a pretty good
approximation what would i do instead
i would i would just do the same thing
riemann did which is have more of them
okay so if i put energy in here
and so on the more trapeziums i get
pcr
the more shapes i get
the closer i'll get the less this um
this error area will be
okay so how do i generalize draw
yourself another graph um just like this
one
and when
now i'm going to go from this is kind of
like basic trapezoidal rule one
trapezium
two function values what happens if we
have more so
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