In this video ...
I want to calculate a
definite integral using
"Reiman Sums"
what I want to do is
I want to calculate
the value of
this definite integral
from 1 to 3 of
x squared
minus x plus 1
d x
and ...
I'm NOT going to do this
by finding an
anti-derivative
if you've seen
this stuff ...
I'm NOT going to do it
this way
I'm going to do it
the long way
using the definition
Let me remind you
on a couple of things here
just in case
and also give you
a couple of formulas
that we need to use here
So the first thing
is simply the definition of
a definite integral
in terms of
Reiman Sums
that's what we have
on the right-hand side
remember to calculate
the value of delta X
we'll let that equal
b minus a over n
so the length of the intergral over n
to calculate the value of
X sub-i star
we'll use the formula
a plus delta x times i
and then there's also
a couple of summation formulas
that you also need to know
if you sum up
the first K natural numbers
we'll get k times n
if we sum up
the first n ... err ...
excuse me, I said that backwards
if k is a constant
and if we sum that up
from 1 to n
we'll get k times n
if we sum up
the first n natural numbers
from 1 to n
the formula is
n times n plus 1 over 2
and ....
if we square them instead
we get this corresponding formula
OK I don't want to talk about
where these come from
that's not my intent
in this video ...
just you want to be able
to use these
so when I go to calculate this stuff
it says my delta X value
for this particular problem
again, b is the upper limit
of integration
so let's write down the original problem
that we had a second ago
that we were trying to calculate
so we want to calculate the value
from 1 to 3
again, of X squared
minus X
plus 1 ...
d x
OK?
so in this case we can calculate
our delta x value
it's going to be the upper limit
minus the lower limit
so that will give us
3 minus 1
over n
with the value
2 over n
so now to get our value
for X sub i
again, we just plug in our 'a' value
the lower limit of integration
is our 'a' value
so we're going to let
X sub i be 'a'
which is 1
we know what delta X is
we just calculated that
that's the value
2 over n
or equivalently
I am just going to write it as
1 plus 2 i over n
so know we know our
X sub i value
and again there is nothing really
to fill in on these bottom formulas
but we'll end up using them
at some point
during the course of this problem
OK so the function
that we're integrating
in this case
this is our f of X
OK so we're trying to integrate
X squared minus X plus 1
so that means our value
f of x sub i star
that's going to equal ...
well originally it was
X squared minus
X plus 1
so if we plug that in we'll get
1 plus 2 i over n quantity squared
minus
1 plus 2 i over n
and I think probably
it's a good idea at this point
to simplify this out
these are pretty long and tedious problems
as you are about to witness
unfortunately
so if I were to foil this out
I believe I would get
1 plus 4 i over n
because I would get
2 i over n plus 2 i over n
that would give me
4 i over n
and then on the last part
I would get
4 i squared over n squared
minus 1
minus 2 i over n
and then I have
my plus 1 hanging out
so the minus 1 and the plus 1
would just cancel out
so we can get rid of those
I've got stuff divided by n
well I've got 4 i over n
minus 2 i over n
that's going to leave me with a
positive 2 i over n after I subtract
so this is the value of our function
f of X sub i star
so now if we plug it in
to the definition
to calculate this thing
let's plug all of this in
so again, we know what our
delta X is ...
that's 2 over n X sub i star
we've calculated our f of X sub i
so it says when we plug it all in
we're going to have the limit
as n goes to infinity
and when we plug in our
f of X sub i
... let's not leave out our summation ...
i equals 1 to n
when we plug in our
f of X sub i star
again, we're going to be calculating
the summation of this
1 plus 2 i over n
plus 4 i squared over n squared
and it says we have to multiply
all of that by
delta X
which is 2 over n
so now we have
re-written our definite integral
in terms of the definition using
Reiman Sums so ...
I think I am going to stop this video here
so this would be the set up ...
look around, this will be 'Part 1'
I'm actually going to compute the sum
in another video
but if you just need
to see how to set it up
that would be
the first part of it
alright, I hope this helps
like I said, if you want to actually
see me compute this
dig around
it should be right nearby