We want to find the surface area of revolution
about the x-axis, for the function
y equals three sine two x,
over the closed interval from zero to pi over two.
Looking at the graph on the lower left,
here's the graph of our function, y equals three sine two x,
over the closed integral from zero to pi over two.
To rotate this about the x-axis,
it would give us this surface,
and our goal is to find the surface area of this surface.
We know from our lesson, since we have a function of x,
we'll be using this formula here to find the surface area,
where the surface area is equal to two pi,
times the integral of r of x, times the square root
of the quantity, one plus f-prime of x squared,
integrated with respect to x, from a to b,
where r of x is the distance from any point on our graph
to the axis of rotation, and we should recognize
this square root as the integrand, when finding arc length.
So, if we select any point on our graph,
let's say this point here,
this distance here would be r of x,
which would be the function value, three sine two x,
and therefore, r of x
equals three sine two x.
F of x is also three sine two x,
and now, we'll find f-prime of x,
which would be three times the derivative of sine two x,
which requires the Chain Rule,
so we'd have three times cosine two x times two,
or six cosine two x,
which means, the surface area
is equal to two pi,
times the integral from zero to pi over two,
of r of x, which is three sine two x,
times the square root
of one plus,
f-prime of x squared.
Well, here's f-prime of x,
so f-prime of x squared would be 36
cosine squared two x,
integrated with respect to x.
Notice how we could factor out this three.
Let's do this, and continue on the next slide.
So, we factor out the three.
That would give us the surface area equals six pi,
times the integral from zero to pi over two
of sine two x
times the square root
of one plus 36, cosine squared, two x.
So, for the next step,
let's evaluate this on the graphing calculator,
and then, we'll do this by hand
using the integration formula, given below,
from the integration table.
Let's first make sure we are in radian mode.
Press the mode key.
Notice how we are in radian mode.
Go back to the home screen.
So, enter six pi,
and we want to select
function integration from the math menu,
so press math, arrow down to option nine,
here, press enter,
and now, we'll enter the integrand, so we have sine two x,
times the square root of one plus 36.
Now, for cosine squared two x,
we'll enter cosine two x, and then, square it,
so in parentheses, you'll have cosine two x.
Close parenthesis for the two x.
Close parenthesis for the cosine, squared,
and then close parenthesis for the square root.
There's the integrand, comma x,
comma zero, comma pi over two.
Close parenthesis, and enter.
So, the approximate surface area is 61.2428.
And this would be square units.
Now, let's also do this by hand using
the integration formula given here, below,
where we have the integral of the square root
of u squared, plus a squared, integrated with respect to u.
So, to begin, let's rewrite this integral
to better fit this form.
And notice how it's also going to require U substitution.
Let's write this as S equals six pi,
times the integral from zero to pi over two, of sine two x.
And let's write this as the square root of,
let's write 36 cosine squared, two x,
as six, cosine two x, squared, plus,
let's write one as one squared.
Notice, in this form, we should be able to recognize
that u is equal to six, cosine two x,
and a equals one.
Notice, this means, differential u is equal to
six times the derivative of cosine two x times dx,
which would give us six times negative sine two x times two,
or negative 12 sine two x, dx.
Notice how the integral contains sine two x, dx,
so let's divide both sides by negative 12,
so this tells us that negative 1/12 du equals
sine two x, dx.
So now, let's write this in terms of u,
so we'll have S equals six pi.
Notice, these limits of integration
are in terms of x, not u, so notice, when x is zero,
u is equal to six times cosine zero, which is six.
And, when x is pi over two,
we have u equals six times cosine,
this would be cosine of pi, which is negative one,
so we have u equals negative six.
Next, sine two x, dx, is equal to negative 1/12 du.
Let's factor out the negative 1/12.
Then we have du, and then we have the square root
of six cosine two x squared.
That would be u squared,
and then, since a is equal to one, we have plus a squared.
And now, we can apply our integration formula give here,
so notice how this simplifies to negative 1/2 pi.
The antiderivative here gives us an extra factor of 1/2,
so we have times 1/2, and then we have
u times the square root of u squared, plus a squared,
plus a squared, natural log,
absolute value of u squared plus a squared,
and we were integrating from six to negative six.
Let's continue on the next slide.
Notice how here we have negative 1/4 pi,
so let's first substitute
negative six for u, then six, and then, find the difference.
Remember, a was equal to one, so a squared is one.
So, when you use negative six, we'd have
negative six times the square root of,
this would be negative six squared, plus one squared,
that's 37,
plus a squared is just one, so we'd have
plus natural log, absolute value of negative six, plus,
again, we have square root 37, and then when u is six,
we have six, and then, the square root of,
six squared plus one squared, that's 37,
plus, again, a squared is one, so we have natural log,
absolute value of six plus square root of 37.
To save some time, I've already evaluated
these expressions on the calculator.
This comes out to negative 1/4 pi, times,
this quantity here is approximately negative 38.98836,
and this quantity here,
is approximately positive 38.98836,
so this comes out to negative 1/4 pi,
times, this difference comes out to negative 77.97672,
so here we have negative pi over four,
times negative 77.97672,
which gives us the same result
of approximately 61.2428.
Then again, this is surface area,
so this would be square units.
I hope you found this helpful.