Surface Integrals, Coordinates of Centers of Gravity, and Moments of Inertia of Surfaces

A surface integral is called an integral of the form $\displaystyle \iint \limits_{\sigma } P \, dy \, dz+Q \, dz \, dx+R \, dx \, dy$. In computing it, the variables $x$, $y$, and $z$ are expressed in terms of two independent variables, using the equation of the surface $f(x,y,z)=0$ or its parametric equations $x=\varphi (u,v)$, $y=\psi (u,v)$, $z=\omega (u,v)$. Another, more explicit way of writing surface integrals is $\displaystyle \iint \limits_{\sigma }(P\cos \alpha +Q\cos \beta +R\cos \gamma ) \, d\sigma$, where $\alpha$, $\beta$, $\gamma$ are the angles of the normal with the coordinate axes; here $d\sigma =\dfrac {dx\, dy}{\cos \gamma }$.

The coordinates of the centers of gravity of parts of homogeneous surfaces having mass are found using surface integrals by the formulas:

Here $S$ is the given part of the surface.

Find the coordinates of the center of gravity of the part of the shell of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ for $x>0$, $y>0$, $z>0$.

Find the coordinates of the center of gravity of the surface of a segment of the same sphere $x^{2}+y^{2}+z^{2}=a^{2}$ for $h<z<a$.

Find the coordinates of the center of gravity of the surface of the helicoid $x=u\cos v$, $y=u\sin v$, $z=hv$ for $0<u<a$, $0<v<\pi$.

Find the coordinates of the center of gravity of the part of the surface $3z=2(x\sqrt{x}+y\sqrt{y})$ bounded by the planes $x=0$, $y=0$, $x+y=1$.

Find the coordinates of the center of gravity of the surface $x^{2}=2c(c-z)$, bounded by the surfaces: $y=kx$, $y=0$, $z=0$.

Find the coordinates of the center of gravity of the part of the surface $x^{2}+y^{2}+z^{2}=a^{2}$ bounded by the surfaces: $x^{2}+y^{2}=ax$, $z=0$.

Find the coordinates of the center of gravity of the part of the surface $x^{2}+y^{2}+z^{2}=a^{2}$ bounded by the planes: $x+y=\pm a$, $x-y=\pm a$.

Find the moments of inertia of the part of the homogeneous surface of the cone $h^{2}(x^{2}+y^{2})=a^{2}z^{2}$ for $0<z<h$ relative to the axis $Oz$.

Find the moment of inertia of the surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ relative to a diameter.

Find the moment of inertia of the surface of the paraboloid $x^{2}+y^{2}=2az$ relative to $Oz$ for $0<z<a$.

Find the moment of inertia of the surface of the spherical segment $x^{2}+y^{2}+z^{2}=a^{2}$ for $h<z<a$ relative to $Oz$.

Compute the double integral $\displaystyle \iint \dfrac {ds}{\rho }$, taken over the surface of the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$, where $\rho$ is the distance from the center of the ellipsoid to the plane tangent to the element $ds$ of the surface of the ellipsoid.

Compute the integral $\displaystyle \iint \dfrac {ds}{\rho^{n}}$, taken over the surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$, if $\rho$ is the distance from the surface element to the point $(0,0,c)$, located outside the sphere.

Show that the integral $\displaystyle w=\iint \limits_{S} \dfrac {\cos (\overrightarrow {r},\overrightarrow {n})}{r^{2}} \, ds=\iint \limits_{S} \dfrac {\partial \frac{1}{r}}{\partial n} \, ds$, taken over the surface $S$, equals the solid angle at which the surface $S$ is seen from the origin. Here $\overrightarrow {r}$ is the radius vector from the origin to the surface element $ds$; $\overrightarrow {n}$ is the normal to the surface, forming an acute angle with the vector $\overrightarrow {r}$ drawn from the origin. The quantity $\dfrac {\partial u}{\partial n}$ is defined by the equality