today we're going to talk about how to find the volume obtained by rotating the region
bounded by these three curves about the line x equals two so we've been given the curves
y equals x cubed y equals zero and x equals one between the three of these they bound
a particular region and we're suppose to rotate that region around the line x equals two and
find the volume that we obtain when we perform that rotation so the first thing we need to
do is realize that we're rotating around a line or around an axis of rotation x equals
two and that's the vertical line x equals two so whenever we're rotating around a line
x equals something it's probably best to solve the equations that we're given here for x
in terms of y so right now we have y equals x cubed and y equals zero we want to solve
those for something in terms of x if we can so for example y equals x cubed if we take
the third root of both sides in other words if we raise both sides to the one third power
then what we'll get is x equals y to the one third and now this equation is solved for
x in terms of y we also have the line y equals zero which we can't really solve for x and
we have the line x equals one so we're in good shape now what we should do next is graph
a picture of these as best as we can so that we have an idea of what we're dealing with
so if we draw here an x y coordinate system the first thing we need to do is graph our
equation so we have x equals y cubed that's the same thing as y equals x cubed and we
know that the graph of y equals x cubed looks something like this so if we graph it like
this it looks something like this and then it would come down here and go off in that
direction but it's basically this shape here so we have y equals x cubed so we'll actually
call this y to the one third and then we have y equals zero so that line y equals zero is
just going to be here y equals zero and we'll say y equals zero there and then we have x
equals one so the line x equals one and we'll say that that's roughly here the line x equals
one so now we've defined and area we have this area here between the three curves and
we're going to rotate this region around the line x equals two so if we have the line here
x equals two we call that the axis of rotation so what we're going to do is we're going to
take this region here in purple and we're going to rotate it around this line so if
we rotated it eventually you know we'd have the region over here like this on this side
and we're going to rotate around the whole thing and it's going to form a volume and
we want to find the value of that volume so what we need to do is realize that we have
intersection points right so our curves intersect each other here and they intersect each other
here we know that the value here is zero zero at the origin this intersection point is at
one one if we didn't know that we could set our curves equal to one another we could set
y to the one third equal to one because they're both equal to x and we would see y to the
one third is equal to one we could cube both sides and we get y equals one so we know that
the value there is one those are our limits of integration those are going to be our values
for a and b so a will be zero and b will be one because our limits of integration we're
going to be integrating on this range here for y from zero to one so let's start looking
at our formula here because we already know a is going to be zero and b is going to be
one what our formula here for the volume tells us is that we have aor here which I've just
abbreviated it's basically axis of rotation so we know that the axis of rotation is the
line x equals two so we're just going to be plugging in two for axis of rotation then
we have outer curve and inner curve so the outer curve is the curve that's farther away
from the axis of rotation and in this case that's the curve y to the one third because
we basically have this distance here between the axis of rotation and the outer curve then
we have the inner curve which is the vertical line x equals one so that distance is right
here between the axis of rotation and the inner curve the curve that's closer to the
axis of rotation the difference between them this distance right here is going to be at
any given point the area of this shape so thats y when we subtract the inner curve from
the outer curve we get the area and if we integrate that and we multiple by pi then
we're going to get the volume of rotation so once you know the outer curve the curve
that's farther away from the axis of rotation and the inner curve the curve that's closer
to the axis of rotation plugging into your formula is simple so we have here the integral
from zero to one because the point where the curves intersect each other at the lower bound
is zero and the upper bound is one so we have zero to one of pi times axis of rotation two
minus the outer curve and the equation of the outer curve is y to the one third so y
to the one third power squared minus pi times axis of rotation minus the inner curve which
we know is x equals one so we just get one there squared and then dy and whenever you're
rotating around a vertical line x equals two so whenever you're rotating around x equals
something you know that you want to plug in values for your outer and inner curve that
are in terms of y so we have y here and we have the line x equals one we know we're going
to have dy and we know that our limits of integration are going to be y values from
zero to one as opposed when you're rotating around a line y equals something and it's
a horizontal line you know that you're going to have x values here you're going to have
you know for example x to the one third and one here you're going to have dx and you're
limits of integration are going to be like you know zero and one here x values along
the x axis but we have the opposite case here where we're rotating about a vertical line
so we have y values and y limits of integration so now it's just a matter of simplifying an
integral as much as we can and evaluating at the limits of integration so we have first
of all we can always factor out pi because we have pi as a constant coefficient on both
of our terms so we'll factor out pi and we'll say zero to one of two minus y to the one
third squared minus two minus one is just one when we square that we just get one and
then dy so now let's go ahead and multiply out this two minus y to the one third squared
so we'll foil it when we multiply two minus y to the one third times two minus y to the
one third we get four minus four y to the one third power plus y to the two thirds two
thirds minus one dy and now we can go ahead and take the integral so we'll get pi times
four y minus four y to the one third we add one to the exponent and we get four thirds
we divide four by four thirds which is the same as multiplying it by three fourths so
we get three y to the four thirds then we have y to the two thirds we add one to the
exponent and we get five thirds we divide by five thirds so we get plus three fifths
y to the five thirds and then minus one just becomes minus y and we're going to be evaluating
that on the limits of integration zero to one now as you know when we're evaluating
on limits of integration we plug in the top number first so we'll plug in one and then
subtract from that whatever we get when we plug in zero so we get pi times and we'll
plug in one four minus or four times one is just one one to the four thirds is still just
one times three is three so we get minus three one to the five thirds is still one times
three fifths is just three fifths and then minus one now we could subtract what we get
when we plug in zero but we'll just get zero zero zero and zero so no sense in writing
all of those out and this is all we're left with we want and go ahead and find a least
common denominator which is going to be five so in order to get a least common denominator
of five we'll multiply each of these other terms by five over five and we'll get twenty
over five minus fifteen over five plus three fifths minus five fifths and when we take
twenty minus fifteen plus three minus five we get a volume of three pi over five and
that is the value of the volume obtained by rotating this region here about the line x
equals two so I hope you found that video helpful if you did like this video down below
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