We're asked to "find the volume formed
by the region enclosed by x equals 9.5y, x equals y cubed,
with y greater than or equal to zero
rotated about the y-axis."
I've already shown the bounded region here
on the coordinate plane,
when we are rotating about the y-axis.
If this region is rotated about the y-axis,
we get the solid shown below.
We will determine the volume, using the washer method,
with rotation about the y-axis where the volume is equal
to pi times the integral from c to d of the square
of big R of y minus the square of little r of y,
integrated with respect to y.
To help us set this up, let's draw a rectangle
that would represent one washer of the volume rotated
about the y-axis.
So the rectangle will be perpendicular
to the axis of rotation.
Let's say, approximately here.
Notice, the thickness of the rectangle is delta y,
which is why we integrate with respect to y.
And big R of y is the outer radius
which would be this length here.
Little r of y is the inner radius,
which would be this length here.
We notice how big R of y is given
by the equation x equals 9.5y.
Little r of y is given by x equals y cubed.
And then for the limits of integration with respect to y,
we integrate from y equals zero to the y-coordinate
of this point of intersection.
So we'd like to find the y-coordinate at this point
of intersection, algebraically.
But let's go ahead and begin to set this up.
We have the volume is equal to pi times the integral.
We'll come back to the limits of integration
where the square of big R of y is the square of 9.5y
minus the square of little r of y is the square of y cubed.
And now we have to determine the limits of integration.
We start integrating at y equals zero.
And again, the upper limit of integration
is the y-coordinate at this point of intersection here.
To find the y-coordinate,
we need to solve the system of equations,
x equals 9.5y and x equals y cubed for y.
So let's work on that below. Let's perform a substitution.
Let's substitute 9.5y for x, in the second equation,
which gives us 9.5y equals y cubed.
Let's set the equation equal to zero
by subtracting 9.5y on both sides,
which gives us zero equals y cubed minus 9.5y.
Let's factor out the common factor of y.
9.5 is not a perfect square, but it is equal to the square
of the square root of 9.5.
So we can factor this as y times y
minus the square root of 9.5
times y plus the square root of 9.5.
So our solutions are y equals zero,
as well as y equals plus or minus the square root of 9.5.
And we can see from the point of intersection it's
in the first quadrant, where y is positive,
and therefore the upper limit
of integration is square root 9.5.
And now let's work on evaluating the integral.
We have the volume is equal to pi times the integral
from zero to the square root of 9.5.
The square of 9.5y is equal to 90.25y squared.
And then we have minus the square
of y cubed is y to the sixth.
And now let's find the antiderivative.
The antiderivative of 90.25y squared is equal
to 90.25 times y cubed divided by three,
or 90.25/3 y cubed
minus the antiderivative of y to the sixth
is y to the seventh divided by seven,
or 1/7 y to the seventh.
Let's continue on the next slide.
So we have pi times big F of the square root of 9.5
minus big F of zero.
So substituting the square root of 9.5,
we have 90.25/3
times the square root of 9.5 cubed.
And then we have minus 1/7 times the seventh power
of the square root of 9.5.
And then for big F of zero, when y is zero,
both quantities are zero.
So let's now go to the calculator.
To four decimal places, we have 1,581.3325,
which does give us the volume of the solid
if the region shown here is rotated about the y-axis.
I hope you found this helpful.