- [Instructor] What we're gonna do in this video
is introduce ourselves to the notion of a definite integral
and with indefinite integrals and derivatives
this is really one of the pillars of calculus
and as we'll see, they're all related
and we'll see that more and more in future videos
and we'll also get a better appreciation
for even where the notation
of a definite integral comes from.
So, let me draw some functions here
and we're actually gonna start thinking about
areas under curves.
So, let me draw a coordinate axes here,
so that's my Y axis, this is my X axis,
actually I'm gonna do two cases.
So, this is my Y axis, this is my X axis
and let's say I have some function here,
so this F of X right over there
and let's say that this is X equals A
and let me draw a line going straight up like that
and let's say that this is X equals B,
just like that
and what we want to do is concern ourselves
with the area under the graph,
under the graph of Y is equal to F of X
and above the X axis and between these two bounds,
between X equals A and X equals B,
so this area right over here
and you can already get an appreciation.
We're not used to finding areas
where one of the boundaries or as we'll see in the future,
many of the boundaries could actually be curves
but that's one of the powers of the definite integral
and one of the powers of integral calculus.
And so, the notation for this area right over here
would be the definite integral
and so, we're gonna have our lower bound at X equals A,
so we'll write it there,
we'll have our upper bound at X equals B,
right over there.
We're taking the area under the curve of F of X,
F of X and then DX.
Now, in the future we're going to,
especially once we start looking at Riemann sums,
we'll get a better understanding
of where this notation comes from.
This actually comes from Leibniz,
one of the founders of calculus.
This is known as the summa symbol
but for the sake of this video
you just need to know what this represents.
This right over here,
this represents the area under F of X
between X equals A and X equals B.
So, this value and this expression
should be the same