In this video we're talking about how to find the area between curves, and more specifically,
about the most difficult part of any area between curves problem, which is how to figure
out which curve is on the top and which is on the bottom, or which curve is on the left
and which is on the right.
So, whenever you have an area between curves problem, there's really kind of three parts
to it.
The first part is finding any points of intersection, so the points where the two curves intersect
each other.
The second part is what I just talked about which is the orientation of the curves.
So, which curve is on the top, and which one's on the bottom, or which curve is on the left,
and which curve is on the right.
And then, once you figure that out, and you have the intersection points, then the third
part of the problem is plugging everything into the area between curves formula, which
is an integral formula, and then evaluating that integral.
So we're really focusing on this second part, because finding the intersection points is
fairly straightforward and easy and then once we've got all this, plugging into the formula
is pretty simple too.
So, the whole area between curves process really hinges on getting this second step
right.
So, how do we go about figuring out which curve is on the top and which is on the bottom?
Well, the first method is just to look at a picture of the graphs.
So, if we are lucky enough to have a picture of the graphs of both curves, then we can
really easily identify which curve is on the top and which one's on the bottom.
So, in this case, the red curve is clearly the top curve, and the blue curve is clearly
the bottom.
And the way that we know that for sure is, we're starting here at the x-axis or at y=0,
and we start moving up, and what do we hit first?
The bottom curve, which is the blue curve, and then if we keep going, we hit the top
curve, or the red curve.
Which makes the blue curve the bottom and the red curve the top.
Now you might be saying, "What about this area out here?"
If we started out here and we move up, we hit the red curve first and then we would
hit the blue curve, which would make the red curve the bottom curve.
But, remember we're looking for the area between the curves, and so, all we care about there
is the area that's between the intersection points.
So our intersection points are here and here, which means the area between the curves that
we're looking for, is the area between about here, x=1 and x=5.
This area in here.
And on this interval, the red curve is clearly the top curve, the blue curve is clearly the
bottom curve.
So if you've got a picture, it'll be really easy to identify which one's the top and which
one's the bottom.
Where things really get tricky is when we don't have a picture of the curves, we've
only been given the equations of the functions themselves.
So, in this example, we've been told we have f(x)=4 and g(x)=(x-2)^2.
But we don't have a picture of the graph of either curve, so all we have to go on is these
equations.
So, when that's the case, how do we figure out which curve is on the top and which one
is on the bottom?
Well, the first thing we have to do is find the intersection points.
We can't determine top and bottom curves without the intersection points, because we need to
know the interval that we're interested in, in order to figure out which curve is top
and which one's bottom.
So, the way that we're going to find the intersection points is by setting the curves equal to each
other.
Realize here that f(x)=4 is the same as saying y=4, and that g(x)=(x-2)^2 is the same as
y=(x-2)^2.
And so because both of these curves are equal to y, we have y on the left hand side of each
one, we can set the right hand sides equal to each other as well.
So we can say (x-2)^2=4.
And then from there we just want to solve for the variable.
So we will take the square root of both sides of the equation.
And the square root will cancel with this squared exponent here, leaving us with just
x-2 on the left.
And then on the right hand side the square root of 4 is going to be +/-2.
Now from here we'll add 2 to both sides and get x=2+/-2.
And so then, we can solve this and say x is going to be equal to either 2-2, which is
0, or 2+2 which is 4.
So x=0 and x=4, those two points are going to turn out to be the intersection points
of f(x)and g(x).
Now that we have the intersection points, what we need to do is pick a value inside
the interval, so in between 0 and 4.
We'll go ahead and pick 2 since it's right in the middle of 0 and 4.
You don't have to pick a value that's directly in the middle.
For example, here we could have picked 1 or 3, anything that's between 0 and 4 will work.
But we'll go ahead and pick x=2.
So x=2 will be kind of our test value.
And what we want to do with that test value is plug it into both of the original equations
f(x) and g(x).
So when we do that, we'll get f(2) is going to be equal to, there's nothing to plug in
here, there's no x variable, so f(2) is still just going to be 4.
But g(2), will give us here when we plug into the right hand side, 2-2 or 0, and then 0^2
is still 0, so g(2) will be 0.
Now this part is really easy.
When we plugged in our test value, which was inside the interval x=0 to x=4.
In other words, between the points of intersection, all we care about at this point is which value
came out larger.
So is 4 bigger or is 0 bigger?
Well 4 is obviously greater than 0, which means f(x) is going to be the top curve and
g(x) is going to be the bottom curve simply because 4>0.
Now we certainly don't need to, but we could graph both of these curves, sketch them, to
confirm the results that we just got.
So remember we just determined that the intersection points were at x=0 and x=4 and when we sketch
both of these curves we confirm that that looks like it's true.
And then between 0 and 4, remember we picked this test value here of 2, between 0 and 4
we determined that f(x) was the top curve and that g(x) was the bottom curve.
Because when we plugged in x=2 to both curves, we got 4 for f(2), and we got 0 for g(2).
Which looks like it's right because g(x) looks like it's value is 0 and f(x) looks like it's
value is 4 at x=2.
Which told us that this curve was the top and this curve was the bottom.
And again, you definitely don't have to sketch the graphs, you can do it all algebraically,
but if you have time to sketch them, then that's a great way to double-check yourself.
And then from there, all we need to do to set up the integral is this.
So we would say the integral from 0 to 4 because our intersection points were at 0 and 4, so
the integral from 0 to 4.
And then all we care about is top curve minus bottom curve.
So what we would say there is top curve f(x) minus the bottom curve g(x), and then we're
integrating with respect to x so we put in a dx.
And if we wanted to plug in here we would just plug in 4 and we would plug in (x-2)^2.
And this would be the integral that gives us the area between curves.
So you can see how finding the intersection points is pretty simple, we just set the curves
equal to each other.
And setting up the integral once we had everything was pretty simple too.
The trickiest part by far was figuring out which curve was on the top and which curve
was on the bottom.
So that'll be our general plan for finding the area between curves, but there are a couple
of things that can happen that will complicate this process a little bit.
So, for example, if we have curves like this, maybe we have one curve here and then one
curve here, we can see that we have this intersection point here.
So if we set these two curves equal to each other and solved for the variable we would
find that there's an intersection point right here.
But sometimes we're going to be asked to find the area between these curves maybe, for example,
over the interval x=1 to x=5.
So we'll sketch those in roughly.
So we may be provided with that interval in the problem itself.
It might tell us "Find the area between these curves over the interval x=1 to x=5."
And we have here 1 and 5.
Well if that's the case, we can't just assume that 1 and 5 are intersection points, even
though they're the endpoints of the interval we were given.
We still have to check for intersection points because we may have one or more intersection
points that occur inside of our interval.
So when that's the case like it is here, that intersection point splits up the interval
1 to 5 into two separate subintervals.
We need to find this area and then also this area here.
So how do we do that?
Well like I said, you'd find the intersection point, let's pretend that this intersection
point here is at x=3.
So you'd find this intersection point, you'd know that you'd be integrating over the interval
1 to 5, so that intersection point is going to break this up into subintervals here.
The subinterval 1 to 3 and the subinterval 3 to 5 right here.
So once you realize that, then just like before, you need to pick a test value on each interval.
So before we just had one continuous area and we picked a test value in that interval.
But here we have two subintervals so we have to pick two test values.
So since this is the interval from 1 to 3 we might pick a test value for this region
of x=2.
And then since this is the interval 3 to 5 we might pick a test value of x=4 for that
interval.
We would plug both of these values into both curves to see which curve returned the higher
value.
So when we plug in x=2 to both the red and the blue curve what we would see is that the
red curve returns a larger value and is therefore the top curve and the blue curve returns a
lower value or a lesser value so it would turn out to be the bottom curve.
And then we would look at our test value of x=4, we would plug that value into both curves,
and we would see that the blue curve returns a larger value and is therefore the top curve
and the red curve returns a lower value and is therefore the bottom curve over these particular
intervals.
And therefore, when we set up the area between curves interval to find this area, we would
set up one integral for each part of the area and then add them together.
So for example, this area on the left, we would say, that's going to be given by the
integral from 1 to 3 since that subinterval is defined by 1 to 3 and then we would say
top curve minus bottom curve which would be red minus blue so red minus blue and then
we're integrating with respect to x so we say dx.
Then we would just add to that the integral from 3 to 5 since that defines that subinterval
and then there we would say the blue curve is the top curve and we would say minus the
red curve which is the bottom curve and again integrating with respect to x.
So the sum of these two integrals put together would define this total area over 1 to 5.
You also can run into an instance where you have multiple intersection points, so for
example, you might have a curve that looks like this and then a curve that looks something
like this.
And as you can see, you have multiple intersection points here.
So again you would just set the curves equal to each other, solve for the variable, and
that would give you all of these intersection points, however many you have.
And then you might be asked to find just the area here on this particular interval, or
maybe for example you're asked to find the area over an interval that looks like this
and so what you would end up with is three distinct pieces of area and so you would need
to find an integral for each piece and then add those three integrals together.
And then you would do the same thing that you did up here, you would define a subinterval
for each piece of area, you would pick a test value inside each interval, you would plug
that into both curves, and then you would see that red is the top blue is the bottom
for the left hand piece, blue is the top red is the bottom for the middle piece, and then
red is the top blue is the bottom again for the piece on the right here.
And then you would put together an integral for each on of those, and add them all together.
So that's what that can look like.
And then the last complicating factor that you can have is when your curves look something
like this.
So you might have a curve here, something like that, and then a curve maybe that looks
like this roughly, and these curves, if you're asked to figure out which curve is the top
curve and which curve is the bottom curve, it gets really tricky because they don't really
look like top and bottom curves.
They look more like left and right hand curves.
And that's actually exactly how you should treat them.
Instead of top bottom you should treat them as left right.
Because in this case the blue curve is definitely to the left of the red curve and that makes
sense because if we start here at the y-axis or at x=0, and we move this way, we hit this
curve first so this curve is on the left and then we keep going and we hit this curve on
the right in between the intersection points here.
When you see curves like this, what's probably going on is that both curves are defined instead
of as y=f(x) and y=g(x), they're defined as x=f(y) and x=g(y).
So they're both defined for x in terms of y and that's why they come out looking like
this.
In that case, you do exactly the same thing that you would for curves defined for y in
terms of x, that turn out to be top bottom curves.
You find the intersection points by setting the curves equal to each other, then you pick
a test value between the intersection points.
So our intersection points are here and here.
So we're looking for a y-value in between the intersection points to be our test value,
and we would take that y-value, that test value, and plug it into both curves.
Whichever curve returns a larger value is the curve on the right, and whichever curve
returns a smaller value is the curve on the left.
And then, once you have that down, you just set up your integral this way.
So your integral would look like this.
You would have the integral and then your limits of integration will be y=a and y=b
where this here is y=a the lower intersection point, and this is y=b the upper intersection
point.
And then you would just take the right curve minus the left curve and then you're just
integrating with respect to y instead.
And that's what your integral will look like for left and right curves.
So those are a couple of the bumps that you might run into, but in general, figuring out
which curve is on the top and which is on the bottom, or which is on the left and which
is on the right, is fairly straightforward and getting comfortable with this can really
help you with your area between curves problems.
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