Find the volume of the solid
generated by turning, through a complete revolution about the 𝑦-axis, the
region bounded by the curve nine 𝑥 minus 𝑦 equals zero and the lines 𝑥 equals
zero, 𝑦 equals negative nine, and 𝑦 equals zero.
Remember, when we rotate a
region bounded by a curve 𝑥 equals some function of 𝑦 and the horizontal lines
𝑦 equals 𝑐 and 𝑦 equals 𝑑 about the 𝑦-axis, we use the formula the definite
integral between 𝑐 and 𝑑 of 𝜋 times 𝑥 squared d𝑦. In our case, the horizontal
lines we’re interested in are given by 𝑦 equals negative nine and 𝑦 equals
zero. So, we’re going to let 𝑐 be
equal to negative nine and 𝑑 be equal to zero. The region we’re interested in
is bounded by the curve nine 𝑥 minus 𝑦 equals zero and 𝑥 equals zero. Now, we said 𝑥 needs to be
some function of 𝑦. So, we make 𝑥 the subject and
we find that our equation is 𝑥 equals 𝑦 over nine. That’s this region. And it looks a little something
like this when we rotate it about the 𝑦-axis.
Substituting everything we know
into our formula for the volume, and we find it’s equal to the definite integral
between negative nine and zero of 𝜋 times 𝑦 over nine squared d𝑦. We take out a constant factor
of 𝜋 and we distribute our parentheses. And we see, our integrand is
now 𝑦 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 𝑦 squared
is 𝑦 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 𝑦-axis is three
𝜋 cubic units.