In school, we have all learnt and memorised a lot of formulas for areas of different geometric
figures, as well as those for volume. And particularly for volumes, most teachers won’t
really tell you where the formulas came from, or just say you will need calculus to understand.
Well sure, it is easy if you know enough about calculus, but calculus is not compulsory in
a lot of math curricula, while the formula for volume of a sphere usually is. This is
basically forcing students to learn without really understanding. Luckily, if we know
Cavalieri’s principle, it is possible to bypass integration, but volume of a sphere
is not the only thing that can be dealt with using Cavalieri’s principle. This video
will mainly focus on finding the volume of a paraboloid, because there are already videos
out there talking about these applications in detail, so I don’t want to repeat here.
Cavalieri’s principle states a very simple fact. Given two different solids with the
same height, we can take horizontal cross sections for each of the solids. In this case,
all of the cross sections of the two solids are the same, so the volumes of the two solids
are the same. More generally, if the two solids have the same cross-sectional areas, then
they have the same volume.
To construct a paraboloid, we first take a part of a parabola. In this illustration,
we have chosen 1 - x^2 as the parabola on the plane y = 0. Rotate this about the vertical
axis to form the paraboloid that we want. The horizontal cross-sections for the paraboloid
are all circles, but they differ in radii, so the cross-sectional areas would not be
the same. More precisely, at any particular height z, the radius of the cross section
would be the length of this line segment, which is the x-coordinate of the corresponding
point on the original parabola. By a very simple manipulation of the formula of the
parabola, we have the x-coordinate to be square root of 1 - z. This means that the cross-sectional
area at height z is pi times the radius squared, which is pi times 1 - z. Or if you prefer,
it is pi times the distance to the tip of the paraboloid. This does not look too promising,
because whenever we change z, the cross-sectional area would change. But here is the trick.
If we flip the paraboloid upside down, and take the cross-section at height z again,
then this time, the area of the cross-section is simply pi times z, because the distance
to the tip of the paraboloid is now z. So if we take a cross section at height z for
both solids, the cross section on the left has area pi times (1-z), and that on the right
has area pi times z. So if we add them together, we would get pi, which doesn’t depend on
z. So we can compare this to a cylinder, with unit radius so that all the horizontal cross-sectional
areas are all just pi, and the cylinder actually circumscribes the original paraboloid.
Basically, if we take the cross section at every possible height, the areas for the two
paraboloids would add up to that for the cylinder, so by Cavalieri’s principle, the sum of
the volumes of the two identical paraboloids is the volume of the circumscribing cylinder.
However, this is just one example of the parabola. We can generalise this argument to all parabolas,
and hence all the paraboloids. The argument is entirely analogous to what we did above,
just with a bit more algebra.
The idea for this video is not actually mine, but from a viewer who sent an email to me,
which you can also do by going to my channel about page, scroll down and see my contact
email there.
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