Question

Evaluate the ∫∫ y dS where S is the portion of the cylinder x2 + y2 = 3 that lies between z = 0 and z = 3.

Answer 1

The cylinder is projected in a circle on the area Oxy. Use the polar coordinate system:

x = p cos ϕ

y = p sin ϕ

|J| = p

0 <= ϕ <= 2 π

Consider:

x² + y² = 3

p² = 3

p = $\sqrt{3}$

0 <= p <= $\sqrt{3}$

$\iint ydS = \iint ydxdy = \int{_0^{2π}} dϕ \int{_0^{ \sqrt{3} }}p∙ p sin ϕ dp = \intop{_0^{2π}} sin ϕ∙$ (p³/3)$\mid{_0^{ \sqrt{3}}}dϕ = \int{_0^{2π}} sin ϕ ∙ (3 \sqrt{3}/3) dϕ = \sqrt{3}∙ \intop{_0^{2π}}sin ϕ dϕ = \sqrt{3} ∙ (-cosϕ ) \mid{_0^{2π}} = - \sqrt{3}∙ (1-1) = 0$