Question

Determine surface area of cylinder by integration

Answer 1

If you'd really want to go all out, let the cylinder represented in cylindrical coordinates (r,θ,z)(r,θ,z)where rr is the radius from the zz-axis, θθ is the azimuthal angle. Now the surface area of a small element of the cylinder will be given by dA=rdθdzdA=rdθdz. We seek to integrate around the cylinder 0≤θ≤2π0≤θ≤2π and 0≤z≤40≤z≤4 with a fixed radius 11. The area of the cylinder is then the integral,

∬AdA=∫40∫2π0dθdz=8π∬AdA=∫04∫02πdθdz=8π

as required.

From basic geometry, the surface area is A=2π⋅4=8πA=2π⋅4=8π.