Introduction to Area Under the Curve Video Lecture | Mathematics (Maths) Class 12 - JEE

let's say we want to find the area under
this function on the interval from 1 to
9 one way to do this is to divide the
region into four segments and to draw a
rectangle inside each segment such at
the upper right corner of each rectangle
touches the function the sum of the area
of these rectangles is an estimate of
the area under the curve the estimate is
obviously poor in this case because the
rectangles just don't fit under the
curve they're too wide we can do better
by doubling the number of rectangles
once twice three times four times five
times and six times until we have a
total of 256 rectangles these narrower
rectangles fit under the curve better
than the wider ones and so the sum of
their areas is a much better estimate of
the area under the curve we could do
this differently by placing rectangles
into the four segments such at the left
edge of each rectangle touches the curve
we could then double the number of
rectangles once twice three times four
times five times and six times the
result is the same a bunch of rectangles
that fit under the curve fairly well and
they provide a decent estimate of the
area under the curve when the area of
all 256 rectangles is summed finally we
could fill the four segments with
rectangles such that the top center of
each rectangle touches the curve
doubling the number of rectangles once
twice three times four times five times
and six times leads to the now-familiar
result that the rectangles better fit
under the curve allowing the sum of the
area of all rectangles to provide a good
estimate of the area under the curve it
does not matter whether we place the
rectangle such that the curve touches
them on the upper right corner in the
center or on the upper left corner so
long as they are narrow enough to fit
nicely under the curve they can provide
an estimate of the area under the curve
what we'd like to do is translate this
concept into math looking at the same
function again let's imagine we want to
find the area under the curve between
these two x-coordinates let's further
imagine that we will estimate the area
using rectangles inside three equally
sized regions to translate this into
math we need labels for the x
coordinates that define the range of
each of these regions from left to right
let's call them x0 x1 x2 and x3 now
let's place a rectangle between x0 and
x1 such at the upper right corner of the
tangle touches the curve at x1 the area
of this rectangle is its height which is
the value of the function at x1 times
its width which is x1 minus x0 to
estimate the area under the curve we
need to add to this the area of the
rectangle in the next region which is
its height the value of the function at
x2 times its width x2 minus x1 finally
we need to add to this the area of the
rectangle in the third region which is
the value of the function at X 3 times
x3 minus x2 let's call this sum of
rectangle areas capital s and give it a
subscript of 3 to indicate the number of
rectangles used in the sum this can be
written more compactly using summation
notation with the variable J serving as
the subscript and taking values from 1
to 3 the argument to the sum is the
height of each rectangle the value of
the function at X sub J times the width
of each rectangle X sub J for the
x-coordinate of the right side of the j3
angle minus the x-coordinate of the left
side of the same rectangle which is X
sub J minus 1 the width can be written
even more compactly as Delta X sub J
with the understanding that this stands
for X sub J minus X sub J minus 1 3
rectangles do not provide a very good
estimate of the area under the curve
from X 0 to X 3 and so we might want to
include more rectangles to do this take
the height of the J 3 angle times its
width and sum from J equals 1 to an
arbitrary number of rectangles in this
is S sub n a sum of the area of n
rectangles and is called a Riemann sum
the number of rectangles could be 6 12
24 or any other number but S sub n only
becomes a good estimate of the area
under the curve if we take the limit of
this quantity as the number of
rectangles the value of n goes to
infinity in the limit of including very
many rectangles S sub n is the area
between the curve and the x-axis from X
0 to X sub N and is written symbolically
as the integral from X 0 to X n of the
function f of X times an infinitesimal
width D X this is the riemann definition
of an integral between two fixed limits
sometimes called a Riemann integral in
the integrand f of X is the height of an
infinitesimally narrow rectangle and DX
is the infinitesimal width of that
tangle as we have learned a Riemann
integral can be interpreted
geometrically as the area between the
curve and the x-axis on an interval but
this is only true if the function lies
above the x-axis
for instance this function lies above
the x-axis between x equals 0 and x
equals one point 185 the Riemann
integral over this interval is positive
one point one to one and it can be
interpreted as the area under the curve
in this interval because the function is
never negative between x equals one
point one eight five and three the
situation is different because the
function is negative the Riemann
integral over this region is negative
two point nine nine seven because the
function is always negative clearly a
real geometric area can never be
negative and so we need to refine our
definition of a Riemann integral a
better definition of a riemann integral
is that it is the net area between the
function and the x axis over some
interval with the understanding that
areas below the x-axis are negative and
areas above the x-axis are positive only
in the case in which the function is
always positive can we interpret the
Riemann integral as a true geometric
area otherwise the value is a net area
so far we have talked about definite
integrals ones that are between fixed
limits and that are written as the
integral from A to B of a function times
a differential width where a and B are
constants these integrals have been
defined in terms of Riemann sums with a
geometric interpretation that is helpful
but that does not provide a practical
way to evaluate integrals this is
provided by the fundamental theorem of
calculus which we'll get to in just a
second if we replace the upper limit B
with the variable X we obtain the
integral from a to X of the function
times a differential width the result is
no longer a value instead the result is
a function of X that we'll call capital
f of X this integral is an indefinite
integral because the upper limit is not
definite it is important to note that
the variable U is just a dummy variable
of integration and could be anything you
t WX and so on the function f of U is
the function f of X with X change to you
for convenience and clarity so that X is
not confused with the upper limit of
integration the fundamental theorem of
calculus states that the derivative of
the function capital f of X which is the
result of an indefinite
integral is equal to the function being
integrated little f of X this defines
differentiation as the inverse of
integration and integration as the
inverse of differentiation this provides
an immediate way to find the indefinite
integral of a function determine what
function when differentiated gives the
function you're trying to integrate for
instance what is the integral of cosine
X the function that when differentiated
gives cosine X is sine X thus sine X
must be the integral of cosine X of
course adding any constants such as C
also gives cosine X when differentiated
we sometimes call this the constant of
integration
and that's what is an integral