Why do we take different elemental thickness for calculation of volume and area of sphere?

Question

Why do we take thickness be differential of distance apart of elemental mass when calculating volume and be differential length of arc when calculating area of the sphere when integrating in terms of angle.

Before going into depth I refer this thread first.

So what I learnt is that when getting small volume we take $$dl = d(R\sin θ) = R\cos θ\cdot dθ <<$$ $$r = R\cosθ$$ $$dV = πr^2\cdot dl$$

While when calculating area we take $dl$, $$dθ = \frac{dl}{R}$$ $$dl = R\cdot dθ <<$$ $$dA = 2πr\cdot dl$$

But like why different elemental thickness ($dl$) for those calculations?

If I wasn't able to make you understand what I said, then see the second figure, the $h$ is taken as $dl$ in calculation of volume while the curved-surface is used as $dl$ for calculation of area

My Question is: WHY

Answer 1

Think of these computations as bases on the piling of cone frustra.

The volume of a single frustrum is the base area by the height, $\pi r^2\,dh$.

The lateral area is the circumference of the base times the slant height, $2\pi r\dfrac{dh}{\cos\theta}$.

(For thin frustra, the difference between the two bases does not matter.)

Answer 2

This is because $\theta \to 0$ so $$ \frac{d \sin \theta}{d\theta}=\lim_{\theta \to 0}\frac{\sin \theta}{\theta}=1 $$ In other words if $\theta$ is small the difference with $\sin \theta$ is much more small.