### Video Transcript

Find the volume of the solid
generated by turning, through a complete revolution about the π¦-axis, the region
bounded by the curve nine π₯ minus π¦ equals zero and the lines π₯ equals zero, π¦
equals negative nine, and π¦ equals zero.

Remember, when we rotate a region
bounded by a curve π₯ equals some function of π¦ and the horizontal lines π¦ equals
π and π¦ equals π about the π¦-axis, we use the formula the definite integral
between π and π of π times π₯ squared dπ¦. In our case, the horizontal lines
weβre interested in are given by π¦ equals negative nine and π¦ equals zero. So, weβre going to let π be equal
to negative nine and π be equal to zero. The region weβre interested in is
bounded by the curve nine π₯ minus π¦ equals zero and π₯ equals zero. Now, we said π₯ needs to be some
function of π¦. So, we make π₯ the subject and we
find that our equation is π₯ equals π¦ over nine. Thatβs this region. And it looks a little something
like this when we rotate it about the π¦-axis.

Substituting everything we know
into our formula for the volume, and we find itβs equal to the definite integral
between negative nine and zero of π times π¦ over nine squared dπ¦. We take out a constant factor of π
and we distribute our parentheses. And we see, our integrand is now π¦
squared over 81. And in fact at this stage, it might
also be sensible to take out this common factor of 81. Then, we know that to integrate a
polynomial term whose exponent is not equal to negative one, we add one to that
exponent and then divide by the new value. So, the integral of π¦ squared is
π¦ cubed over three.

Then, our volume is π over 81
times zero cubed over three minus negative nine cubed over three. And, of course, zero cubed over
three is zero. We might then choose to write
negative nine as negative nine times negative nine squared or negative nine times
81. And this means we can now simplify
by dividing through by 81. And then, negative negative nine
divided by three is just three. And so, we found that the volume of
the solid generated by rotating our region about the π¦-axis is three π cubic
units.