how to get approximation using differentials?

Question

Approximate the volume of a hemispherical shell with thickness 0.15 cm, diameter 10 cm, and height 5 cm using differentials.

Answer 1

volume of a sphere = (4/3)pi(r^3)

volume of a hemisphere = volume of half a sphere = (4/3)pi(r^3)/2 = (2/3)pi(r^3)

for r= 5 cm, the hemisphere's volume = (2/3)pi(5^3) = 250pi/3

for the shell, r=5 to r=5-.15 = 4.85

V= (2/3)pi(4.85)^3 = 228.17pi/3

Volume of the shell = 250pi/3- 228.17pi/3 = about 7.3pi cm^3 = about (21.83)(3.14159)/3 = about 22.86 cm^3

if you calculate the volume with differentials, you should get close to that number, about 23 cm^3

the method for calculating the formula for the volume of a sphere involves differentials. Look up the proof of the volume formula