标量曲面积分

Weighting the surface

Mass of a spherical shell

Average value on a surface

Choosing the surface for the problem

To integrate $f(x,y,z) = z$ over a surface parametrized by $\mathbf{X}(u,v)$, what replaces $z$ in the integrand?

In $\iint_S z^2\, dS$ over the unit sphere, the integrand is $\cos^2\varphi \sin\varphi$. What supplies the $\sin\varphi$?

The average of $x^2$ over the unit sphere is asserted to be $1/4$. What is the fastest refutation?

A hemisphere of radius $2$ has constant density $3$ per unit area. Give its mass in the form a*pi.

A surface of area $12$ carries $\iint_S f\, dS = 36$. Give the average value of $f$, as a plain number.