Linear Approximation estimate

Question

Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.02 cm thick to a hemispherical dome with a diameter of 40 meters.

I don't understand this one.

Answer 1

ΔV = 2/3π(r + Δr)³ - 2/3πr³ ≈ dV = d/dr(2/3πr³)dr = 2πr²Δr = 2π·2000²·0.02 cm³ = 5026548 cm³