For both volume & surface area, the sphere is split into many discs and the area or circumference of the discs are summed up in an integral. But the summation process uses $dy$ for volume & $r\,d\theta$ (arc-length) for surface area. Why this discrepancy?
Supposing we have a sphere in the $x$-$y$-$z$ plane where you split the sphere into discs along the $y$ axis.. If you visualise the problem from $z$ axis looking down over the $x$-$y$ plane.. The sphere will look like a circle and the disc will be a line segment inside the circle (chord). The length of the line segment will be the diameter of the disc. And the point where the line segment and circle meet - (x,y) can be solved by plugging in the value of y and the x we solve for will then be the radius of the disc.
Now to calculate surface area, we need to sum up the circumference of each disc $ s(x) = 2\pi x$ & and for volume, we need to sum up the area of each disc $ v(x) = \pi x^2 $
Say, the point $(x,y)$ makes an angle $\theta$ with the origin. Then for surface area, we assume for length $r\,d\theta$, the disc radius is not changing (across arc length) & we integrate it as: $$\int s(x)\, rd\theta $$
But for volume, instead of using the arc length, we use the diameter $dy$ to integrate it as: $$\int v(x) \,dy$$
Why this discrepancy? In both cases, the number of discs is the same so why should the summation be different?
I tried interchanging the summation process and when i converted everything into polar co-ordinates ($x = r\,cos\theta, y = r\,sin\theta $) i get an extra $cos\theta$ since $ dy = rd\theta.cos\theta$
The same happens to me when i calculate Moment of Inertia for a solid sphere & hollow sphere. Similarly when i calculate gravity for a point outside a solid sphere & hollow sphere.
Can someone please tell me, why we need to change the summation process?? What decides the summation process, why the difference?