Surface Integrals

By how much do the integrals $I_{1}=\displaystyle \iint_{S} (x^{2}+y^{2}+z^{2})\, dS$ and $I_{2}=\displaystyle \iint_{P} (x^{2}+y^{2}+z^{2})\, dP$ differ (where $S$ is the surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$, and $P$ is the surface of the octahedron $\left|x\right|+\left|y\right|+\left|z\right|=a$ inscribed in this sphere)?

Compute $\displaystyle \iint_{S} z\, dS$, where $S$ is the part of the surface $x^{2}+z^{2}=2az\ (a>0)$ cut off by the surface $z=\sqrt{x^{2}+y^{2}}$.

Compute the following surface integral of the first type: $\displaystyle \iint_{S} (x+y+z)\, dS$, where $S$ is the surface $x^{2}+y^{2}+z^{2}=a^{2}$, $z\geqslant 0$.

Compute the following surface integral of the first type: $\displaystyle \iint_{S} (x^{2}+y^{2})\, dS$, where $S$ is the boundary of the solid $\sqrt{x^{2}+y^{2}}\leqslant z\leqslant 1$.

Compute $\displaystyle \iint_{S} \dfrac {dS}{(1+x+y)^{2}}$, where $S$ is the boundary of the tetrahedron $x+y+z\leqslant 1$, $x\geqslant 0$, $y\geqslant 0$, $z\geqslant 0$.

Compute $\displaystyle \iint_{S} \left|xyz\right|\, dS$, where $S$ is the part of the surface $z=x^{2}+y^{2}$ cut off by the plane $z=1$.

Compute $\displaystyle \iint_{S} \dfrac {dS}{\rho }$, where $S$ is the surface of an ellipsoid, and $\rho$ is the distance from the center of the ellipsoid to the plane tangent to the ellipsoid's surface at the element $dS$.

Compute $\displaystyle \iint_{S} z\, dS$, where $S$ is the part of the helicoid $x=u\cos v$, $y=u\sin v$, $z=v$ $(0\leqslant u\leqslant a,\ 0\leqslant v\leqslant 2\pi )$.

Compute $\displaystyle \iint_{S} z^{2}\, dS$, where $S$ is the part of the conical surface $x=r\cos \varphi \sin \alpha$, $y=r\sin \varphi \sin \alpha$, $z=r\cos \alpha$ $(0\leqslant r\leqslant a,\ 0\leqslant \varphi \leqslant 2\pi )$, and $\alpha$ is a constant $\left(0<\alpha \leqslant \dfrac {\pi }{2}\right)$.

Compute $\displaystyle \iint_{S} (xy+yz+zx)\, dS$, where $S$ is the part of the conical surface $z=\sqrt{x^{2}+y^{2}}$ cut out by the surface $x^{2}+y^{2}=2ax$.

Prove Poisson's formula

where $S$ is the surface of the sphere $x^{2}+y^{2}+z^{2}=1$.

Find the mass of the paraboloidal shell

whose density varies according to the law $\rho =z$.

Find the moment of inertia, about the $Oz$ axis, of the uniform spherical shell of density $\rho_{0}$

Find the moment of inertia, about the line

of the uniform conical shell of density $\rho_{0}$

Find the coordinates of the centroid of the part of the uniform surface $z=\sqrt{x^{2}+y^{2}}$ cut out by the surface $x^{2}+y^{2}=ax$.

Find the coordinates of the centroid of the uniform surface $z=\sqrt{a^{2}-x^{2}-y^{2}}$ $(x\geqslant 0,\ y\geqslant 0,\ x+y\leqslant a)$.

With what force does the uniform truncated conical surface of density $\rho_{0}$

attract a point mass $m$ located at the vertex of the surface?

Find the potential, at the point $M_{0}(x_{0},y_{0},z_{0})$, of the uniform spherical shell of density $\rho_{0}$, $x^{2}+y^{2}+z^{2}=a^{2}$, that is, compute the integral

where

Compute

where

Sketch the graph of the function $u=F(t)$.

Compute the integral

where

Compute the integral

where $S$ is the moving sphere

and it is assumed that $\sqrt{x^{2}+y^{2}+z^{2}}\geqslant a\geqslant 0$,

Compute the following surface integral of the second kind: $\displaystyle \iint_{S} x\, dy\, dz+y\, dx\, dz+z\, dx\, dy$, where $S$ is the outer surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$.

Compute the following surface integral of the second kind: $\displaystyle \iint_{S} f(x)\, dy\, dz+g(y)\, dx\, dz+h(z)\, dx\, dy$, where $f(x)$, $g(y)$, $h(z)$ are continuous functions and $S$ is the outer surface of the rectangular parallelepiped $0<x<a$, $0<y<b$, $0<z<c$.

Compute the following surface integral of the second kind: $\displaystyle \iint_{S} (y-z)\, dy\, dz+(z-x)\, dx\, dz+(x-y)\, dx\, dy$, where $S$ is the outer surface of the conical surface $x^{2}+y^{2}=z^{2}$ $(0\leqslant z\leqslant h)$.

Compute the following surface integral of the second kind: $\displaystyle \iint_{S} \dfrac {dy\, dz}{x}+\dfrac {dx\, dz}{y}+\dfrac {dx\, dy}{z}$, where $S$ is the outer surface of the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$.

Compute the following surface integral of the second kind: $\displaystyle \iint_{S} x^{2}\, dy\, dz+y^{2}\, dx\, dz+z^{2}\, dx\, dy$, where $S$ is the outer surface of the spherical shell $(x-a)^{2}+(y-b)^{2}+(z-c)^{2}=R^{2}$.