Riemann Sums, Trapezoidal
Rule, and Simpson’s Rule
Haley Scruggs
1st Period
3/7/11
A Brief Overview of what Riemann
Sums, Trapezoidal Rule and Simpson’s
Rule do…
• Each of these method estimate the area of a
curve using rectangles.
• As your number of rectangles increase so does
the accuracy of the area.
A Refresher On Left Riemann Sums
• The (b-a) is the interval in which the area estimation
is being calculated.
• N is the number of rectangles
• B is the higher number in the interval and A is the
lower.
Left Riemann Sum Cont.
• To get your different x’s, you must first add
to the A until you get B. Note that the number of
rectangles you are given is the number of times
you must add
.
• After getting your different x’s you then plug
them back into the original equation and then
use the Riemann Sum Equation to find your
estimated area.
Lets See an Example
* Note that in Left Riemann Sum you use the first number in
the interval but not the last when finding the area. This is
because you are coming from the left side of the curve not the
right*
Example Cont.
* Plug these numbers
back into the original
equation*
Next Up on the Refresh Train is Right
Riemann Sums
• The only real difference in the equation for
Right Riemann Sum is the fact that you are
going to use the last number in the interval
but not the first. This is because you are
coming from the right side of the curve
instead of the left.
Lets see that Example
*Remember you
are not going to
use the first
number in the
interval but you
will use the second
number in the
interval when
calculating your
estimated area. *
Example Cont.
Finally the last Riemann Sum is
Midpoint
• There is a slight difference between Midpoint
Riemann Sum and the other two Riemann
Sum techniques.
• The first is that you use both of the numbers
in the interval to find your final answer.
• The other difference is that you divide the x’s
by two and then plug that into the f(x)
equation to find your area.
Now for the Example
*After you add your ¼
to all the numbers, go
back and add the
number you added the
¼ to and the new
number together and
divide by two then plug
that number into the
equation.*
These are the numbers
you plug into the
equation.
Example Cont.
Alright next up is Trapezoidal
• Some slight differences in the Trapezoidal Rule
equation to notice.
• First when finding the area you now have a 2n
instead of the original n.
• The other thing to notice is that you have to
multiply the f(x) by 2, except for the first and last
f(x).
Trapezoidal Cont
• Note that even though the multiplier out front
is
you still will use
to find what your
x’s.
• The number of rectangles is still used as a
check to make sure your math is correct when
finding your x’s, too.
• You also use all of the x’s.
The Trapezoidal Ex
There is more than one way to write the
equation. The A is the lower number on
the integration sign and the B is the higher
number on the integration sign.
You do not multiply
the N by 2 to find your
additive.
Example Cont.
*Note the
new
multiplier on
the outside.*
*You have to
multiply f(x) by 2
now except for
on the first and
last term.*
Finally on the Review list is Simpson’s
Rule
There are three differences between Simpson’s
Rule and the Trapezoidal Rule.
Simpson’s Cont.
1. There is now a 3n on the bottom instead of
a 2n.
2. You have to alternate between 2 times and 4
times the f(x) instead of the constant 2 times
f(x).
3. Simpson’s Rule can only be used for an even
number of rectangles.
Simpson’s Example
You do not multiply
the N by 3 to find your
additive.
As can see the problem is started off the same way as all
the others you will see the differences in later steps
Example Cont.
Now for the easier way to do these
problems… The Calculator Method!
• There are two calculator methods.
• The first will give you Trapezoidal and all the
Riemann Sums .
• The second will give you Simpson’s Rule.
Now for the Steps on how to do it
• Turn the calculator on and go
to Programs
• Scroll down until you see
Riemann and hit enter then hit
enter again.
• Once there enter your
function and press enter, enter
your interval and hit enter,
and finally enter your number
of rectangles and hit enter.
• Once it is done graphing your
function, hit enter and choose
which method you would like
it to use and hit enter.
• It should pop out an answer.
•Turn on the calculator and go to Y=
and put in your equation.
•Then go to Programs .
•Scroll down till you see SIMP then
hit enter.
•Put in the number of rectangles and
hit enter.
•Put in what A = and hit enter
•Put in what B= and hit enter
•It should spit out an answer based
on the equation you put in your y=.
Calculator Try Me
Left Riemann
Simpson’s Rule
Right Riemann
Trapezoidal Rule
Midpoint Riemann
n=4
[0,2]
Answers:
Left Riemann
=.32835
Right Riemann
=.2601
Midpoint =1.375
Simpson’s =
3.14029
Trapezoidal=
1.98352
Now for the Try Me Problems
Left Riemann
Try Me: Left Riemann Answer
Try Me: Right Riemann
Right Riemann Answer
Try Me: Midpoint
Midpoint Answer
Trapezoidal Try Me
Trapezoidal Answer
Simpson Try Me
Simpson Answer
One Last Thing… An FRQ
• 2004 (Form B) AB3
T(min) 0
5
10
15
20
25
30
35
40
V(t)
7
(mpm)
9.2
9.5
7
4.5
2.4
2.4
4.3
7.3
A. Use a midpoint Riemann sum with four subintervals of equal length and values
From the table to approximate
Show the computations that lead to your
answer.
=229
© Haley Scruggs 2011