Volume and surface area of the sphere using integration

[hamalyas]

Homework Statement

i was trying to find the surface area of the sphere using integration, ( by revolving circle on the x-axis )

the thing is it doesn't work as the volume problem. i mean in volume problem to get the volume of the sphere, you would start with circle and start slice it into little pieces, then you would multiply the area of that slice with the thickness which is dx

but in the surface problem you would multiply the circumference of the slice with the incremental arch Length which is ds

why i cannot in the surface problem multiply the circumference by the thickness dx, and i have to multiply it by the arc length ds

actually i uploaded a pdf file to clarify things

Homework Equations

The Attempt at a Solution

 

[hamalyas]

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