Mass, centre of mass and moments of material curves and surfaces

$x / a+y / b = 1, x \geqslant 0, y \geqslant 0$;

$x^{2}+y^{2} = a^{2}, y \geqslant 0$;

$y^{2} = 2 x, 0 \leqslant x \leqslant 2 ;$

$y^{2}-p^{2} = 2 p x, x \leqslant 0$

$x^{2} / a^{2}+y^{2} / b^{2} = 1, y \geqslant 0, a > b$

$y = a \operatorname {ch}(x / a), 0 \leqslant x \leqslant a$.

$x = a \sin t, y = b \cos t, 0 \leqslant t \leqslant \pi / 2, a > b$;

$x = a \sin^{3} t, y = b \cos^{3} t, 0 \leqslant t \leqslant \pi / 2$;

$x = a(t-\sin t), y = a(1-\cos t), 0 \leqslant x \leqslant 2 \pi$.

$r = 2 a \cos \varphi , 0 \leqslant \varphi \leqslant \pi / 2$;

$r = a(1+\cos \varphi ),-\pi \leqslant \varphi \leqslant \pi$;

$r = a e^{\varphi }, 0 \leqslant \varphi \leqslant 2 \pi$.

$x = R \cos \varphi , y = R \sin \varphi ,\left|\varphi \right| \leqslant \alpha \leqslant \pi$;

$x^{2 / 3}+y^{2 / 3} = a^{2 / 3}, x \geqslant 0, y \geqslant 0$;

$y = a \operatorname {ch}(x / a),\left|x\right| \leqslant b$;

$x = y^{2} / 4-(\ln y) / 2,1 \leqslant y \leqslant 2$;

$x = a(t-\sin t), y = a(1-\cos t), 0 \leqslant t \leqslant 2 \pi$;

$r = a(1+\cos \varphi ), 0 \leqslant \varphi \leqslant \pi$;

$r = a e^{\varphi }, \pi / 2 \leqslant \varphi \leqslant \pi$.

$y = e^{x}, 0 \leqslant x \leqslant 1 / 2$

$x = R \cos \varphi , y = R \sin \varphi , 0 \leqslant \varphi \leqslant \alpha \leqslant 2 \pi$;

$x^{2}+(y-a)^{2} = R^{2}, a > R$.

its lateral surface;

its total surface (assuming they are homogeneous).

the lateral surface;

the total surface.

a spherical zone of height $h$;

a segment with base radius $R$ and height $R \sqrt{3} / 2$, cut off from a paraboloid of revolution by a plane perpendicular to its axis.

$x^{2}+y^{2} = R^{2}$;

$x^{2 / 3}+y^{2 / 3} = R^{2 / 3}$.

the lateral surface of a cylinder with radius $R$ and height $h$, relative to its axis;

the lateral surface of a cone with height $h$ and base radius $R$, relative to its axis;

a sphere of radius $R$, relative to its diameter.

$\Gamma$ — the segment $A B, A(1 ; 1), B(2 ; 3) ; \rho (x ; y) = 2 x+y$;

$\Gamma$ — the segment $A B, A(1 ; 0), B(4 ; 6) ; \rho (x ; y) = \sqrt{y+2} / x$;

$\Gamma : y = x^{2} / 2, A(1 ; 1,5), B(2 ; 2) ; \rho (x ; y) = y / x$;

$\Gamma : y^{2} = x, A(1 ; 1), B(4 ; 2) ; \rho (x ; y) = y$;

$\Gamma : y = 2 x^{3 / 2} / 3, A(0 ; 0), B(4 ; 16 / 3) ; \rho = k s$, where $s$ is the arc length from the point $(0 ; 0)$.

$\Gamma : r = a \sqrt{\cos 2 \varphi } ; \rho = k r$;

$\Gamma : r = a(1+\cos \varphi ) ; \rho = k \sqrt{r}$;

$\Gamma : x = a(t-\sin t), y = a(1-\cos t), 0 \leqslant t \leqslant 2 \pi ; \rho = y^{3 / 2}$;

$\Gamma : x = a \cos^{3} t, y = a \sin^{3} t, 0 \leqslant t \leqslant \pi / 2 ; \rho = \sqrt[3]{y}$;

$\Gamma : x = \ln \left(1+t^{2}\right), y = 2 \operatorname {arctg} t-t, 0 \leqslant t \leqslant 1 ; \rho = y e^{-x}$;

$\Gamma : x^{2} / a^{2}+y^{2} / b^{2} = 1, x \geqslant 0, y \geqslant 0 ; a > b ; \rho = y$;

$\Gamma : x^{2 / 3}+y^{2 / 3} = a^{2 / 3} ; \rho = \left|x y\right|$;

$\Gamma : x^{2}+y^{2} = a x ; \rho = \sqrt{x^{2}+y^{2}}$.

$\Gamma : x = \cos t, y = \sin t, z = t, 0 \leqslant t \leqslant 2 \pi ; \rho = \left(x^{2}+y^{2}+z^{2}\right)^{-1}$;

$\Gamma : x = a t, y = a t^{2} / 2, z = a t^{3} / 3,0 \leqslant t \leqslant 1 ; \rho = \sqrt{2 y / a}$;

$\Gamma : x = t \cos t, y = t \sin t, z = t, 0 \leqslant t \leqslant 2 \pi ; \rho = \sqrt{x^{2}+y^{2}+z^{2}}$;

$\Gamma : x = a e^{t} \cos t, y = a e^{t} \sin t, z = a e^{t},-\infty < t \leqslant 0 ; \rho = k z$;

$\Gamma$ — the arc of the curve $y = x^{2} / \sqrt{2}, z = x^{3} / 3$ with initial point $A(0 ; 0 ; 0)$ and terminal point $B(4 ; 8 \sqrt{2} ; 64 / 3) ; \rho = k \sqrt{x^{2}+y^{2}}$;

$\Gamma$ — the arc of the curve $y^{2}-4 x^{2} = 3 z^{2}, y^{2} = x, z \geqslant 0$, with initial point $A(0 ; 0 ; 0)$ and terminal point $B(1 / 4 ; 1 / 2 ; 0) ; \rho = z$;

$\Gamma = \left\{ x^{2}+y^{2}+z^{2} = a^{2}, x+y+z = a\right\} , \rho = x^{2}$.

$\Gamma : y = a \operatorname {ch}(x / a),\left|x\right| \leqslant a$;

$\Gamma : x = a(t-\sin t), y = a(1-\cos t), 0 \leqslant t \leqslant 2 \pi$;

$\Gamma$ — the arc of the circle $r = R,\left|\varphi \right| \leqslant \varphi_{0} \leqslant \pi$;

$\Gamma$ — the cardioid $r = a(1+\cos \varphi )$;

$\Gamma : x^{2 / 3}+y^{2 / 3} = a^{2 / 3}, y \geqslant 0$;

$\Gamma : \sqrt{x}+\sqrt{y} = \sqrt{a}$;

$\Gamma : y^{2} = x^{2} / 3+x^{3}, x \geqslant 0$.

$\rho = \rho_{0} =$ const;

$\rho = \rho_{0} e^{-z / h}$, take $\varphi_{0} = 2 \pi n, n \in N$.

$\Gamma :\left|x\right|+\left|y\right| = a$;

$\Gamma : x^{2 / 3}+y^{2 / 3} = a^{2 / 3}$;

$\Gamma : x = a(\cos t+t \sin t), y = a(\sin t-t \cos t), 0 \leqslant t \leqslant 2 \pi$.

over the surface of the cube $0 \leqslant x \leqslant a, 0 \leqslant y \leqslant a, 0 \leqslant z \leqslant a$ with surface density $\rho = \rho_{0} x y z$;

over the sphere $x^{2}+y^{2}+z^{2} = R^{2}$ with density: а) $\rho = \rho_{0} \sqrt{x^{2}+y^{2}}$, б) $\rho = \rho_{0}\left(x^{2}+y^{2}\right)$;

over part of the elliptic paraboloid $x^{2}+y^{2} = 2 z, z \leqslant 1$ with density $\rho = \rho_{0} z$;

over part of the hyperbolic paraboloid $x^{2}-y^{2} = 2 z$, cut out by the cylinder $x^{2}+y^{2} = 1$, with density $\rho = \rho_{0}\left|z\right|, \rho_{0} =$ const.

$x+y+z = a, x \geqslant 0, y \geqslant 0, z \geqslant 0$;

$x^{2}+y^{2}+z^{2} = R^{2}, z \geqslant 0$.

$\rho = \rho_{0}$;

$\rho = \rho_{0} \sqrt{x^{2}+y^{2}}$;

$\rho = \rho_{0}\left(x^{2}+y^{2}\right), \rho_{0} =$ const.

$x^{2}+y^{2}+z^{2} = R^{2}, x \geqslant 0, y \geqslant 0, z \geqslant 0$

$z = \sqrt{R^{2}-x^{2}-y^{2}}, x \geqslant 0, y \geqslant 0, x+y \leqslant R$;

$z = \sqrt{x^{2}+y^{2}}, x^{2}+y^{2} \leqslant x$;

$z = 2-\left(x^{2}+y^{2}\right) / 2, z \geqslant 0$;

$x = u \cos v, y = u \sin v, z = v, u \in [0 ; 1], v \in [0 ; \pi ]$.

$x+y+z = 1, x \geqslant 0, y \geqslant 0, z \geqslant 0$

$z = \frac{h}{r} \sqrt{x^{2}+y^{2}}, x^{2}+y^{2} \leqslant r^{2}$.

$x^{2}+y^{2} = 2 a z, z \leqslant a$, relative to the axis $O z$;

$x^{2} / a^{2}+y^{2} / a^{2} = z^{2} / b^{2}, 0 \leqslant z \leqslant b$, relative to the line $y = 0$, $z = b$.

$x^{2} / a^{2}+y^{2} / a^{2}-z^{2} / c^{2} = 0,\left|z\right| \leqslant c ;$

$z^{2}-x^{2}-y^{2} = a^{2},\left|z\right| \leqslant a \sqrt{2}$.