You plan to paint a sphere of radius 200 cm with a coat of pain .2 cm thick. Use differentials to approximate the amount of paint that is needed. Use the equation for the volume of a sphere.

Answer 1

v = 4/3 πr^3
dv = 4πr^2 dr
so, dv = 4π*200^2*.2 = 32000π cm^3

note that the dv is just the surface area times the thickness, as you'd expect for tiny thicknesses.