PROOF 159

Prove.

The surface area of a sphere

In space.

What you may use

A1the sphere has a surface area S(ρ)
the area of the sphere of radius ρ
A2S(ρ)t ≤ ΔV ≤ S(ρ+t)t
squeeze for the shell volume — the volume of a shell of thickness t is pinched by the areas of the two spheres
A3V(ρ) = ⁴⁄₃πρ³
volume of the sphere (Proof 158)
A4(xⁿ)′ = nxⁿ⁻¹
derivative of a power (Proof 135)
A5(cf)′ = c f′
derivative of a constant multiple
A6f′(a)=lim(f(a+s)−f(a))/s
definition of the derivative — s → 0, from both sides
A7f ≤ c ⟹ lim f ≤ c
limits respect inequalities — likewise in the other direction
A8x≤y, y≤x ⟹ x = y
antisymmetry of the order
A9t>0: x ≤ y ⟹ x/t ≤ y/t
dividing an inequality by a positive number
A10non-overlapping pieces add
additivity of volume
A11associativity, commutativity, cancellation
arithmetic of the reals
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

Write V(ρ) for the volume of the ball with center O and radius ρ, and S(ρ) for the area of the sphere bounding that ball.

Proof.