Surface integrals of the first kind (with respect to area)

Let $S = \left\{ (x,y,z) \in \mathbb {R}^{3} \mid x^2+y^2+z^2=1\right\}$ denote the unit sphere in $\mathbb {R}^{3}$. Evaluate the surface integral over $S$:

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 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.

Prove the equality

where

and the integration is taken over the surface of a sphere of radius $r$ centered at the point $(x,y,z)$.

Compute the integrals. $\iint_{S}(x+y+z) d S$, where:

Compute the integrals. $\iint_{S}\left(x^{2}+y^{2}\right) d S$, where:

Compute the integrals. $\iint_{S}\left(x^{2}+y^{2}+z^{2}\right) d S$, where:

Compute the integrals. $\iint \frac{d S}{(1+x+y)^{2}}, S$ — the surface of the tetrahedron $x+y+z \leqslant 1, x \geqslant 0, y \geqslant 0, z \geqslant 0$.

Compute the integrals.

Compute the integrals.

Compute the integrals.

Calculate the integrals.

Calculate the integrals. $\iint_{S}\left(x^{2}+y^{2}+(z-a)^{2}\right)^{-n / 2} d S, \quad n \in N, \quad S$ — the sphere $x^{2}+y^{2} + z^{2} = R^{2}$.

Calculate the integrals. $\iint_{S} z^{2} d S, S$ — part of the conical surface $x = u \cos v \sin \alpha , \quad y = u \sin v \sin \alpha , \quad z = u \cos \alpha$,

$\alpha = \mathrm{const}, \alpha \in (0 ; \pi / 2)$, singled out by the conditions $u \in [0 ; 1], v \in [0 ; 2 \pi ]$.

Calculate the integrals. $\iint_{S} z d S, S$ is the surface

Calculate the integrals. $\iint_{S} f(r) d S$, where $r = \sqrt{x^{2}+y^{2}+z^{2}}, \quad f(r) = \begin{cases} 1-r^{2}, & r \leqslant 1, \\ 0, & r \geqslant 1, \end{cases} S$ is the plane $x+y+z = a$.

Calculate the integrals. $\iint_{S} f(r ; z) d S$, where $r = \sqrt{x^{2}+y^{2}}, \quad f(r ; z) = \begin{cases} r^{2}, & r \leqslant z, \\ 0, & r \geqslant z, \end{cases} \quad S-$ the sphere $x^{2}+y^{2}+z^{2} = R^{2}$.