Why do I get two different expression for $dV$ by different methods?

Question

So, I was taught that if we have to find the component for a very small change in volume say $dV$ then it is equal to the product of total surface of the object say $s$ and the small thickness say $dr$.

For example let us take a sphere of radius $r$, then by this method, $$dV = 4πr^2dr.$$

Now, if we use calculus, then, $$V = \frac{4}{3}πr^3$$ $$dV = 4πr^2dr$$

Now both the expressions are equal and this the same case for a cylinder as well, but for a cube of side length say $r$ the expression by the method comes out to be, $$dV = 6r^2dr.$$

Now by the calculus method, $$ V = r^3$$ $$ dV = 3r^2dr.$$

Now, we have two different expressions for $dV$ however this should not be possible in maths as both the expressions are not equal to each other.

Am I doing something wrong here or am I missing something? And if I ever have to take the component in physics should I always go with the earlier method and not the calculus approach?

Answer 1

To make it easier imagine doing this for a cuboid instead, where we're just growing one of the three side lengths not all three.

Imagine we have a cuboid with side lengths $(a,b,c)$ and we grow $a\mapsto a+da$. This gives you an infinitesimal slab of side lengths $b$ and $c$ but we get one of these slabs at each end of the side we grew and each one of these two has thickness $da/2$.

Going back to the original case of imagining growing a cube from side length $r$ to $r+dr$ we get 6 infinitesimal slabs but each is only of thickness $dr/2$.

Alternatively doing the calculus properly avoids this mistake.

Answer 2

You are trying to differentiate from first principles and failing with the cube.

Multiply this expression out and take the limit.
$\delta V = (r+\delta r)^3 -r^3 \to 3\, r^2 \,\delta r$ to the first order in $\delta r$ as $\delta r \to 0$.

You can do a similar thing for the sphere with one variable, the radius.

Note that for the cylinder there are two independent variables, the length and the radius, and I think that you happened to vary the length.

Following on from the cylinder, the cube has the three independent variables of a cuboid amalgamated into one variable, the length of each of the sides, as is the case for a sphere which is a particular example of an ellipsoid.