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## Main Question or Discussion Point

I've always thought of dxat the end of an integral as a "full stop" or something to tell me what variable I'm integrating with respect to.

I looked up the derivation of the formula for volume of a sphere, and here, dx is taken as an infinitesimally small change which is

Now I'm really confused. Is it correct to think of it this way?

Part two to my question:

Using this same logic of using infinitesimals, I tried to find the surface area of a sphere and looked at it as the sum of infinite rings.

[tex]\displaystyle A = \int_0^r 2\pi x dy[/tex]

[tex]=> \displaystyle A = \int_0^r 2\pi \sqrt{r^2-y^2} dy[/tex]

I looked up the derivation of the formula for volume of a sphere, and here, dx is taken as an infinitesimally small change which is

**multiplied**by the area of a disc(pi r^2) giving [tex]\displaystyle V = 2\pi \int_0^r x^2 dy[/tex] which is the sum of these infinitesimals.Now I'm really confused. Is it correct to think of it this way?

**Is there any other way to prove this result without using infinitesimals?**Also, if I'm integrating from 0 to r, wouldn't this give me the area of only half the sphere?Part two to my question:

Using this same logic of using infinitesimals, I tried to find the surface area of a sphere and looked at it as the sum of infinite rings.

[tex]\displaystyle A = \int_0^r 2\pi x dy[/tex]

[tex]=> \displaystyle A = \int_0^r 2\pi \sqrt{r^2-y^2} dy[/tex]

**But this is wrong. Why?**