Video Transcript
Consider the region between the
curves 𝑦 equals five 𝑥 squared and 𝑥 squared plus 𝑦 squared equals two, for 𝑦
greater than or equal to zero. Find the volume of the solid of
revolution obtained by rotating this region about the 𝑥-axis, giving your answer to
two decimal places.
In this example, we’re looking to
find the volume of a solid formed by rotating the area between two curves about the
𝑥-axis. So let’s begin by sketching this
area. 𝑦 equals five 𝑥 squared is a
quadratic. So its graph is a parabola with its
lowest point at zero. And the function 𝑥 squared plus 𝑦
squared equals two is the equation of a circle centered at zero with radius root
two.
The region specified is for
𝑦-values greater than or equal to zero. So, in fact, we’re only concerned
with an area above the 𝑥-axis. And the region to be rotated about
the 𝑥-axis to form a solid is the shaded region between the two curves as shown in
the diagram. The solid obtained by rotating this
region might look a bit like a doughnut, but with an indent rather than a hole at
its center.
To find the volume of this solid of
revolution, we recall that the volume 𝑉 of a solid generated by revolving the
region bounded by two curves, 𝑦 equals 𝑓 of 𝑥 and 𝑦 equals 𝑔 of 𝑥, where 𝑓 is
greater than or equal to 𝑔 on the interval 𝑎, 𝑏, about the 𝑥-axis is given by 𝑉
equals 𝜋 times the definite integral with respect to 𝑥 from 𝑥 equals 𝑎 to 𝑥
equals 𝑏 of 𝑓 of 𝑥 squared minus 𝑔 of 𝑥 squared.
In our case, within the specified
region, the circle with radius root two is greater than five 𝑥 squared apart from
at the points of intersection, where they’re equal. And what we’re actually doing in
our integration is summing the areas of the infinitely many cross-sectional vertical
disks or washers in between where the two curves meet.
So, now for the function 𝑓, if we
rearrange 𝑥 squared plus 𝑦 squared equals two to make 𝑦 the subject, we have 𝑦
equals the square root of two minus 𝑥 squared, which is 𝑓 of 𝑥. 𝑔 of 𝑥 is then five 𝑥
squared.
Now, the limits of integration 𝑎
and 𝑏 are the 𝑥-values where the two curves intersect. And we can find these values in
various ways, perhaps using a calculator or graphing software. We’ll do this here by equating our
two functions and solving for 𝑥. This means we need to solve 25𝑥 to
the fourth power plus 𝑥 squared minus two equals zero for 𝑥. Substituting 𝑢 equals 𝑥 squared,
we can solve the resulting quadratic for 𝑢. And now making some space for our
calculation, using the quadratic formula or otherwise, we have 𝑢 equal to negative
one plus or minus the square root of 201 all over 50.
Now, since this is equal to 𝑥
squared, we can only use the positive result. And evaluating this, we find 𝑥
squared equals 0.26354 and so on. Next, taking the square root on
both sides, we have 𝑥 equal to positive or negative 0.51337 to five decimal
places. So these values are our limits of
integration 𝑎 and 𝑏. And with our functions 𝑓 of 𝑥 and
𝑔 of 𝑥, we have the integral as shown.
To make things a little easier to
manage, we can note that our solid is symmetric about 𝑥 is zero and the
𝑦𝑧-plane. So we can take our lower bound to
be zero and double the result. Now, taking the squares of our
functions, our integrand is two minus 𝑥 squared minus 25𝑥 to the fourth power. Integrating each term separately,
we then have two 𝜋 times two 𝑥 minus 𝑥 cubed over three minus 25 over five 𝑥 to
the fifth power, which is five 𝑥 to the fifth power, all evaluated between 𝑥 is
zero and 0.51337.
Substituting our limits of
integration in and evaluating, where for 𝑥 is equal to zero each term is zero, and
we restrict the value of 𝑏 to two decimal places for reasons of space, we have two
𝜋 times 0.80335 and so on. This evaluates to 5.0476 and so
on. And so the volume of the solid of
revolution formed by rotating the given region about the 𝑥-axis is 5.05 cubic units
to two decimal places.
