Area and volume calculation (no integration))

[Jhenrique]

I can compute the area of the rectangle formed by Δx and Δy simply by product ΔxΔy.

Now, how can I to compute the area in gray given Δr and Δθ?

Also, I can to compute the volume of a parallelepiped formed by Δx, Δy and Δz, simply multiplicand ΔxΔyΔz. But, how can I compute the volume in shpherical coordinates formed by Δρ, Δφ and Δθ?

Convention:

 

[jedishrfu]

If you know the radius then you can compute the wedge by computing the area of circles r and r+dr and an angle ratio dtheta/(2*pi) for dtheta measured in radians:

area(wedge) = area(circle(r+dr) - area(circle(r)) * (dtheta / 2*pi)

In spherical coordinates you have a more complicated situation where you could derive a formula for the 3D wedge using calculus.

If you're thinking the ancient greeks didn't know calculus so how did they algebraically arrive at the answer then
you need to read about their calculus precursor the method of exhaustion:

http://en.wikipedia.org/wiki/Method_of_exhaustion

http://www.math.ubc.ca/~cass/courses/m446-03/exhaustion.pdf

http://www-groups.dcs.st-and.ac.uk/history/HistTopics/The_rise_of_calculus.html