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- Hydraulics: Rotating Vessel
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- General Solution of $y' = x \, \ln x$
- engineering economics: construct the cash flow diagram
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- Maxima and minima (trapezoidal gutter)
- Special products and factoring
- Newton's Law of Cooling
- Law of cosines
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Re: zone of a sphere
I think integration will do, I did not try it though. Here is the derivation of total surface area of the sphere by integration: http://www.mathalino.com/reviewer/derivation-formulas/derivation-formul….
It is interesting to note that the area of spherical zone, $A = 2\pi rh$ will become area of the sphere if $h = 2r$.