today we're going to be talking about how to find the volume obtained by rotating these
two curves about the line y equals one so essentially what we've been asked to do is
find the intersection points of these two curves y equals x squared and x equals y squared
and therefore define the region enclosed by these curves and then imagine that that region
is rotated around the line or around the axis y equals one and find the volume of that region
so before we go into the formula we're going to use let's go ahead and draw a picture of
this so that we understand what we're trying to do we're going to zoom in on the area between
x equals zero and x equals one so we're going to draw an x y coordinate plane and call this
x equals one and y equals one and go ahead and sketch these curves so we can see what
we're talking about so before we do that it's probably easiest to solve both of these equations
in terms of y so we have already y equals x squared but if we solve this second equation
here x equals y squared if we solve that for y by taking the square root of both sides
what we'll get here is y equals square root of x and now we have two curves that are defined
for y in terms of x so if we can graph roughly y equals x squared we know that that's just
a parabola that opens up where the vertex at zero zero so we can graph that we know
it has one point at zero zero and another point at one one and so we can graph roughly
this parabola what it would look like maybe like this in this section then we have the
curve y equals the square root of x which is the opposite we know that it also has a
point at zero zero and it also has a point one one but it goes the opposite way it looks
roughly like this so we have these two curves and we can say that this one is the square
root of x and that the other one we drew is x squared and you can see that they intersect
each other at zero zero and one one if we didn't know what both of these curves look
like we could easily set these equal to each other and solve for x so we could just say
x squared is equal to square root of x square both sides and we get x to the fourth is equal
to x and we can see that this equation is only true when x equals zero and x equals
one so we would know that the intersection points of these curves are x equals zero and
x equals one so what we know now that is because they intersect each other at these two points
we have this region that is enclosed by or bounded by these two curves and that's this
region in here what we're going to do is take this region and pretend that we rotate it
around this line here so we have the line y equals one like this and we're going to
take this region and we're going to rotate around the line so essentially we're going
to rotate it around the line and that will form a volume and we want to find the value
for that volume so let's turn now to our formula and we have several things going on here in
our formula but it's not as complicated as it initially looks the formula is for the
volume is the integral from a to b and a and b are limits of integration and they are the
x values of our intersection points so we know that our curves intersect at zero and
they intersect at one so that means that our limits of integration are going to be from
zero to one and then we have pi times and aor I just abbreviated basically it's the
axis of rotation minus the outer curve minus pi times the axis of rotation minus the inner
curve and we're squaring both of these values notice so the axis of rotation is the axis
that we're rotating around the line that we're rotating around and we know that that's y
equals one so the value for axis of rotation here will just be one the outer curve is the
curve that is farther away from the axis of rotation and the inner curve is the one that's
closer to the axis of rotation so what we can see here is that our outer curve is the
curve x squared and so when we say axis of rotation minus the outer curve the value that
we're finding is the distance here from the axis of rotation to this outer curve right
because the axis of rotation is one if we wanted to find the area or the distance from
the axis of rotation to the x axis that would just be one because we have the line y equals
one so from here all the way to the x axis we'd have one if we subtract the outer curve
here we'll just get the distance between the axis of rotation and this outer curve so we
have that distance then if we look at the axis of rotation minus the inner curve that
gives us the distance here and so when we subtract them from one another we get the
distance between them basically from here all the way down to here at any given time
which is the area between the curves so that's why we have the formula set up that way so
really it's just a matter of plugging in the values that we already know so we have the
integral from zero to one because our limits of integration and our points of intersection
are between the x values zero and one here so we have that times pi minus our axis of
rotation is at y equals one so we have one minus our outer curve and we know that that's
x squared so we say x squared and part of our formula is to square that value so we
say squared minus pi times axis rotation one minus our inner curve which we know to be
square root of x and we square that as well dx then once we have things plugged into our
integral formula like this it's just a matter of simplifying what's inside the integral
and then evaluating the integral so lets go ahead and do that so the first thing we can
do is notice that both terms inside of our integral are multiplied by pi so that's a
constant coefficient that we can factor outside of the integral so we'll get pi times the
integral from zero to one of one minus x squared squared minus one minus square root of x squared
dx now we just need to multiply out what's inside the integral here so we'll get zero
to one if we take one minus x squared times one minus x squared we'll get one minus two
x squared plus x to the fourth minus if we multiply one minus square root of x times
one minus square root of x we'll get one minus two square root of x plus square root of x
times square root of x is just x and then dx so we'll continue with our simplification
and we'll get one minus two x squared plus x to the fourth we'll distribute this negative
sign so we'll get one or minus one plus two times square root of x minus x dx now before
we take the integral let's just go ahead and cancel some things and reorder our terms so
we have one minus one those two are going to go away we'll end up with pi times the
integral from zero to one of and we'll reorder our terms from highest degree to lowest degree
so x to the fourth minus two x squared minus x plus two square root of x dx and now that
we've simplified as much as possible we can go ahead and take the integral so we'll get
pi times one fifth x to the fifth minus two thirds x cubed minus one half x squared keep
in mind that this two times the square root of x is essentially two times x to the one
half so if we call this if we call this two times x to the one half we add one to the
exponent so one half plus two halves gives us three halves so we'll get plus x to the
three halves and then we want to divide two by three halves two divided by three halves
is the same thing as two times two thirds the reciprocal so we'll get four thirds here
out in front and we're going to be evaluating this whole thing on the range zero to one
now remember that when we're evaluating a definite integral we always plug in the top
number here and then subtract whatever we get when we plug in the bottom number so we'll
get pi times plug in our top number first one so we'll get one fifth times one to the
fifth power is just one fifth minus two thirds times one to the third power is just minus
two thirds minus one half times one squared is just minus one half plus four thirds times
one to the three halves is just plus four thirds and then we would subtract and plug
in zero but obviously we'll get zero zero zero and zero when we plug that in so we don't
need to subtract all of those zeros we can just close it off there now we can either
use our calculator or better yet just go ahead and find a common denominator we have a least
common denominator here of thirty so we'll get pi times and we'll find a common denominator
for thirty for each of these so to get the first one to have a denominator of thirty
we'll have to multiply by six over six so we'll get six over thirty here we'll multiply
by ten over ten and we'll get twenty over thirty here we'll multiply by fifteen over
fifteen and we'll get minus fifteen over thirty and here we'll multiply by ten over ten so
we'll get plus forty over thirty and when we take six minus twenty minus fifteen plus
forty we'll get eleven over thirty so times eleven over thirty which means that our final
answer is eleven pi over thirty which is equal to the volume so the volume obtained by rotating
this region here all of the area inside of this region enclosed by these curves when
we take that area and we rotate around this line y equals one to get a volume the volume
is eleven pi over thirty so I hope you found that video helpful if you did like this video
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