We want to find the area of the surface obtained
by rotating the curve y equals two x cubed
from x equals zero to x equals two about the x axis.
So looking at this first graph,
here we have y equals two x cubed.
We want to consider this graph on the interval
from zero to two,
here's x equals two.
And then rotate this about the x axis,
and then find the area of the surface.
So if we rotate this curve of the interval
from zero to two about the x axis,
we'd have this blue surface here.
And our goal is to find the area of this surface.
Looking at our notes below,
notice how we have two formulas here.
This first is for the surface area of revolution
for any horizontal axis.
And here we have a second formula
that's only about rotation about the x axis.
Let's first look at the most general formula.
We have the surface area equals
the integral from a to b of r of x,
times the square root of the quantity
one plus the square of f prime of x
integrated with respect to x.
Where r of x is the distance from the curve
to the axis of rotation,
which we can also think of the radius function.
Notice how the only difference between this formula
and the formula for rotation about the x axis,
is that r of x is equal to the function f of x.
So going back over to our graph here,
notice how the distance from the curve to the x axis
is this distance here,
which would be r of x,
which in our case though,
because we have rotation about the x axis
is the same as f of x.
So notice how the given function is y equals two x cubed,
so for our formula f of x equals two x cubed.
We'll also have to find f prime of x.
F prime of x is equal to six x squared.
Which means the surface area is equal to
two pi times the integral from zero to two,
of f of x which is two x cubed,
times the square root of the quantity
one plus the square of f prime of x,
which would be the square of six x squared
integrated with respect to x.
Let's go ahead and simplify this.
Let's first factor out the two.
So we have four pi
times the integral from zero to two of x cubed.
And let's write the square root as a rational exponent,
so we'd have the quantity one plus,
this would be 36 x to the fourth,
raised to the 1/2 d x.
Now let's evaluate this on the next slide.
Notice how this requires u substitution
where we'd have u equals one plus 36 x to the fourth.
So differential u is equal to the derivative
of one plus 36 x to the fourth times d x,
which would be zero plus,
this would be 144 x to the third d x.
Let's go ahead and divide both sides by 144.
So we have 1/144 d u equals x cubed d x.
So I can think of all of this as
again x cubed d x is equal to 1/144 d u.
And this would be u to the 1/2.
So we integrate, we're going to have four pi,
times
we'll have 1/144 with respect to u
we'd have u to the 3/2 divided by 3/2
or 2/3 u to the 3/2.
So with respect to x, we'd have times 2/3
times the quantity one plus 36 x to the fourth
raised to the power of 3/2.
Let's go ahead and clean this up.
Four pi times 1/144 times 2/3 is equal to
pi divided by 54.
So we're left with the quantity one plus 36 x to the fourth,
raised to the power of 3/2,
limits of integration are still from zero to two.
Now when we define big f of b minus big f of a,
so we have pi divided by 54 times,
one x is two, we'd have one plus 36 times two to the fourth
all raised to 3/2,
minus one x is zero,
we just have one of the 3/2.
So simplifying.
We have pi divided by 54 times
this ends up being 577 raised to 3/2
minus one to the 3/2 is just one.
So this would be the exact surface area
which would be square units.
Let's also get a decimal approximation.
We have pi divided by 54,
times the quantity 577 raised to the power of 3/2
or three divided by two,
right arrow,
minus one,
closed parenthesis
and enter.
So to four decimal places,
this is approximately 806.2848 square units.
So we found the exact and approximate surface area
of this surface here,
formed by rotating y equals two x cubed
of the interval from zero to two about the x axis.
I hope you found this helpful.