We're actually going to use
the shell method or tubes method
to determine the volume of solid
formed by rotating the region bounded by
f of x equals negative three sine x,
y equals zero, x equals pi, and x equals two pi
about the y-axis.
Looking at this graph in the lower left,
we should recognize that f of x equals
negative three sine x,
would be the sine function with an amplitude of three,
reflected across the x-axis,
which would be this graph here.
Y equals zero is the x-axis,
x equals pi is the vertical line here,
passing through pi on the x-axis,
and x equals two pi is this vertical line,
passing through two pi on the vertical axis.
So this shaded region is the bounded region
rotating about the y-axis, which would be here.
So if we rotate this region about the y-axis,
it would give us this solid here.
So our goal is to use the shell or tubes method
to determine the volume of this solid.
And the shell or tubes method
is the best choice to find this volume
because we have a function of x
and a vertical axis of rotation.
And to determine the volume using the shell method,
the vertical axis of rotation, we use this formula here.
To help us set up the integral,
let's draw a representative rectangle
which represent one shell or one tube of the volume.
When using the shell method,
we always draw the rectangle parallel to
the axis of rotation.
So this should be one representative rectangle,
which if rotated about the y-axis,
would give us one shell or one tube of the volume.
Now, looking at our formula, the volume is equal to
two pi times the integral from a to b
of rho of x times h of x,
integrated with respect to x,
where rho of x is the radius function,
and h of x is the height function.
So using the representative rectangle,
the radius function would give us the distance
from the rectangle to the axis of rotation,
which would be this distance here.
This is a horizontal distance equal to x,
and therefore rho of x is equal to x.
The height of the rectangle would be this distance here.
Notice how this height would be given by
the y value of the function,
and because y equals f of x,
the height function, h of x,
is equal to negative three sine x.
This is all the information we need
in order to find the volume of the solid.
So the volume, v, is equal to
two pi times the integral.
The limits of integration will be from pi to two pi
and the integrand function is rho of x times h of x,
which is x times negative three sine x.
Integrated with respect to x.
So we have dx.
Let's go ahead and factor out the negative three
and write this as negative six pi
times the integral from pi to two pi
of x, sine x.
Let's evaluate this on the next slide.
In order to evaluate this depth integral,
we have to use the technique of integration by parts.
And in our class, we have not learned
integration by parts yet.
So we are going to evaluate this
using the graphing calculator.
If you're watching and you're in a different class
where you've already learned integration by parts,
you can refer to the notes here below
to review how to set up the integral
using integration by parts.
But again, for our class,
we'll be using the T84 to evaluate this integral
and determine the volume.
From the home screen,
we'll first enter negative six pi.
To access pi we press second,
and then the exponent key, or caret, here.
To access the depth integral,
we press math, and then nine.
The lower limit of integration is pi.
Once we enter pi, we press the right arrow,
which takes us to the upper limit of integration,
which is two pi.
Right arrow to enter the integrand function, which is x,
sine x,
close parenthesis, right arrow.
The variable of integration is x,
so we enter x for dx, and then enter.
And our homework system asks us
to enter the exact value,
or the value entered to four decimal places.
So the decimal approximation would be 177.6529 cubic units.
If you do try to find the exact value
using integration by parts,
the exact value is
18 pi squared cubic units.
Going back to the first slide,
we now know if we rotate this region about the y-axis,
it would give us this solid, and this solid
has a volume of approximately 177.6529 cubic units.
I hope you found this helpful.