Computation of Moments. Coordinates of the Centroid

Find the static moment and moment of inertia of a semicircular arc of radius $a$ with respect to the diameter through its two endpoints.

Find the static moment and moment of inertia of a uniform triangular plate with base $b$ and height $h$ with respect to its base ($\rho =1$).

Find the moment of inertia of a uniform elliptical plate with semi-axes $a$ and $b$ with respect to its major axis ($\rho =1$).

Find the static moment and moment of inertia of a uniform cone with base radius $r$ and height $h$ with respect to its base plane ($\rho =1$).

Prove Pappus's first theorem: the area of the surface of revolution generated by rotating an arc $C$ about an axis that does not intersect it equals the product of the length of the arc and the length of the circle traced out by the centroid of the arc.

Prove Pappus's second theorem: for a solid of revolution generated by rotating an area $S$ about an axis that does not intersect it, its volume equals the product of the area $S$ and the length of the circle traced out by the centroid of that area.

Find the coordinates of the centroid of the circular arc: $x=a\cos \varphi$, $y=a\sin \varphi$ ($\left|\varphi \right|\leqslant \alpha \leqslant \pi$).

Find the coordinates of the centroid of the area enclosed by the parabolas: $ax=y^{2}$, $ay=x^{2}$ ($a>0$).

Find the coordinates of the centroid of the area

Find the coordinates of the centroid of a uniform hemisphere of radius $a$.

Find the coordinates of the centroid $C(\varphi_{0},r_{0})$ of the arc $OP$ from the point $O(-\infty ,0)$ to the point $P(\varphi ,r)$ on the logarithmic spiral

As the point $P$ moves, what curve does the point $C$ trace out?

Find the coordinates of the centroid of the area enclosed by the curve $r=a(1+\cos \varphi )$.

Find the coordinates of the centroid of the area enclosed by the first arch of the cycloid $x=a(t-\sin t)$, $y=a(1-\cos t)$ ($0\leqslant t\leqslant 2\pi$) and the $Ox$ axis.

Find the coordinates of the centroid of the solid of revolution obtained by rotating the area $0\leqslant x\leqslant a$, $y^{2}\leqslant 2px$ about the $Ox$ axis.

Find the coordinates of the centroid of the hemisphere $x^{2}+y^{2}+z^{2}=a^{2}$ ($z\geqslant 0$).