Finding Area of the Region bounded by a Curve and a Line Video Lecture | Mathematics (Maths) for JEE Main & Advanced

hi well in this tutorial what I want to
do is introduce you to the way we find
out areas under graphs that is areas
bounded by two lines x equals a and x
equals B the x-axis and the curve y
equals f of X and to do that it can be
shown that the area this shaded area
here is given by the integral from A to
B of f of X with respect to X and to
illustrate this what I want to do is run
through a typical question where we've
got to find the area bounded by the
curve y equals x squared plus 1 the
lines x equals minus 1 and x equals 3
and the x-axis
now if you get a question like this and
a graph is not drawn what I would always
suggest is a sketch can often bring out
the question all right so let's just
start by drawing a sketch so we need our
y-axis we'll have that as our y-axis and
x-axis and we've got the curve y equals
x squared plus 1 if it was y equals x
squared
it'd be a parabola going through the
origin but we're adding one to that so
it's a parabola that is going to go
through one on the y axis coming up like
this okay so that's the graph then of y
equals x squared plus 1 now we've got to
find the area from x equals minus 1 to x
equals 3 now x equals minus 1 will be a
line say down here where X is minus 1 we
go across to 3 so let's just say that
that's 3 and we're after the area
bounded by the curve these two lines in
the x axis that's that area inside
between these two blue lines let's just
shade that ok so we can see it let's
just do that and that okay
so we've got our area
now to work out that area then let's
just put it here the area is going to
equal according to the formula the
integral of our curve which is x squared
plus 1 x squared plus 1 we've got a
couple of terms here so we need to put
that in brackets we integrate that with
respect to X and the limits go from
minus 1 to 3 so 4 minus 1 to 3 and in
the usual way when we get something like
this we integrate each term so if we
integrate the first term x squared add 1
to the power and divide by the new power
and for the constant plus 1 that's just
going to be 1x or just simply X and we
put this in square brackets with the
limits going from minus 1 to 3 now we
all we need to do is substitute 3 in for
the X and then subtract what we get when
we substitute the minus 1 in for X so if
we do that we've got 3 cubed all divided
by 3 plus another 3 let's put that in
square brackets and then we subtract
what we get when we substitute the minus
1 in so that'd be minus 1 or cubed all
divided by 3 plus -1 and we'll finish
that square bracket off there working
out the first bracket we've got 27
divided by 3 so that's going to be 9 9 +
3 is 12 and then we've got - and then
this is minus 1/3 minus another one so
that's minus 4/3 so we've got 12 minus
minus 4/3 12 plus 4/3 and that comes out
as 40 over 3 now we're dealing with an
area here so I feel it's a good idea to
put down square units there just to
indicate that you're working out an area
ok well that brings us to the end of
this particular
a video on finding an area bounded by a
curve and the x-axis and a couple of
lines here but in other tutorials what
I'm going to show you is how we cope
with areas when the area is below the
x-axis we have to treat it in a slightly
different way so hopefully you'll look
at that video
you
you