Formulas of Surface Area - Applications of integrals

S = ∫ 2Πyds          rotation about x-axis

S = ∫ 2Πxds          rotation about y-axis

Where,

ds = √ 1 + (1+ (dy / dx)²) dx                  

if y = f (x), a < x < b

ds = √ 1 + (1+ (dx / dy)²) dy                  

if y = h (y), c < y < d

There are some things to note about these formulas. Very firstly, notice that the variable in the integral itself is all time the opposite variable from the one we are rotating about.  2^nd, we are allowed to make use of either ds in either formula.  The meaning of this is that there are, in some way, four formulas here.  We will select the ds based on which is the most suitable for a specified function and problem.