hey guys this is Eddie - magic monk today we're going to talk about how to
find the area under a curve using integration so the goal of this lesson
is let's say we have a curve on the Cartesian plane so this is the x-axis
this is the y-axis let's say we have a curve like this one and we wanted to
find the area from the y-axis all the way to let's say somewhere here let's
say we want it to find out this shaded area here how we can do that using
integration okay so now that's gonna be a bit complicated to start off with that
so let's just start off with a very simple graph so let's say we have a
Cartesian plane X on the horizontal Y on the vertical and let's draw a very
simple straight line equation where we have y equals x right just a very simple
straight line graph and I'm gonna label the axes as well so the x-axis let's say
we have one two three vertical axes I also have one two three and you can see
that let's say I wanted to find the area
under the curve well in this case a straight line between X is y
to X is 3 so let's say I wanted to find this shaded area and well let's not use
any integration techniques at the moment let's just find the area using a simple
area of a triangle rule so the area of a triangle is simply base times height
divided by 2 isn't it you know that's a pretty simple math
from I don't know jr maths so you should be able to know that so area of a
triangle is the base in this case is 3
centimeters or millimeters or whatever units you want to define it as 3 and the
height of the triangle is also 3 so it's 3 times 3 divided by 2 which is 9
divided by 2 which is 4.5 now that's finding the area without any
integration techniques now firstly what we want to do is integrate this equation
right so to find the area under the curve in this case a straight line to
find this area the first thing we want is to integrate the equation so our
original equation is X so let's find the integral of X and because we want to
find out the area from X 0 to X is 3 we're gonna put these two coordinates 0
and 3 next the integral sign so I'm going to put
zero down the bottom and three at the top and now this becomes the definite
integral now what this means is we're going to find the area under the curve
of the equation X between position 0 and position 3 for X so the definite
integral from 0 to 3 of X DX is equal to and now we're just going to integrate
the equation in the middle and if you remember how to integrate we just write
x squared over 2 using the integration formula and now we're going to write 3
at the top and 0 at the bottom
now what that means is I'm gonna substitute the number 3 first into the
equation so I'm gonna substitute 3 squared over 2 and then I'm gonna - and
I'm gonna substitute the number 0 into the formula 0 squared over 2 so now what
does this become 3 squared over 2 is 9 over 2 - 0 squared over 2 is just 0 so
the answer is 4.5 again so now we have just found the area under the curve so
we found this area here using integration now you might ask what how
does this method work I mean how is it that if you integrate something that it
gives you the formula for the area under a curve or if
you want to see a full proof of that it's probably you should probably watch
another video on it and I think I'll make that video soon but usually in high
school you wouldn't be asked to provide a proof as to how an integral of an
equation gives you the area under the curve so you don't have to worry about
the proof in the test but if you're interested in finding out how it works I
should be making a video on it sometime in the future but for now just
understand that if you integrate an equation you end up with the formula
that gives you the area under the curve so that's pretty useful so now let's try
to integrate a more complicated equation to find the area under the curve
okay so I've just thrown a very simple graph in geogebra a very simple curve
and let's say I want to find the area under the curve between X is 1 and X is
3 so I want to know this whole area here
right so what we would do now is we will integrate from so definite definite
integral from X is 1 to 3 of negative x squared plus 4x minus 1 DX all right so
we're gonna find the integral of that and let's use the integration formula so
we have negative x cubed over 3 plus 4x squared over 2 minus X and we're gonna
put the number 3 at the top one at the bottom so we're gonna substitute
number 3 into the formula first so we have negative 3 cubes and notice how
when our the formula is set up that way we're gonna cube it first before we put
the negative on it over 3 plus 4 times 3 squared over 2 minus 3 minus and then
now we substitute the number 1 into it so negative 1 cubed over 3 plus 4 times
1 squared over 2 minus 1
okay so now let's let's try to do it by hand first instead of typing it all in
the calculator so we have negative 27 over 3 plus 4 times 9 is 36 over 2 minus
3 minus 1 cubed is 1 negative 1 over 3 plus 4 over 2 minus 1 so this is all now
just year five math I guess so we have negative 9 plus 18 minus 3 minus just
going to type it in the calculator now
and the answer is 5 and 1/3 so if you have a look at our graph and you can
sort of mentally check whether it makes sense by just looking at the graph
because let's say we divide up this into a rectangle ask where in a triangle so
let's say we divide this area up into a little squid oops let me make it dark up
so let's say I divide this area up into a square here so then we have 2 by 2
which is 4 units squared for let's say centimeters and then this little bit
here let's say we convert it into a little triangle all right so we have 2
on the bottom so we have two centimetres on the bottom
the height is one centimeter so 2x1 is two divided by two is one so it is about
five and the extra bit the five and a third that could come from this little
bit that wasn't included in my triangle all right so you sort of got a mentally
check whether the area makes sense okay well thanks guys for watching the
tutorial on how to find the area under a curve this is for solving the most basic
problem only there will be other problems where you are you might need to
use some more complicated techniques but we'll talk about that later thanks for
watching see you guys next time