Computing the surface area of a solid of revolution

Find the area of the surface formed by rotating the following curve: $y=x\sqrt{\dfrac {x}{a}}$ ($0\leqslant x\leqslant a$) about the $Ox$ axis.

Find the area of the surface formed by rotating the following curve: $y=a\cos \dfrac {\pi x}{2b}$ ($\left|x\right|\leqslant b$) about the $Ox$ axis.

Find the area of the surface formed by rotating the following curve: $y=\operatorname {tg}x\left(0\leqslant x\leqslant \dfrac {\pi }{4}\right)$ about the $Ox$ axis.

Find the area of the surface formed by rotating the following curve: $y^{2}=2px$ ($0\leqslant x\leqslant x_{0}$):

Find the area of the surface formed by rotating the following curve: $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ ($0<b\leqslant a$):

Find the area of the surface formed by rotating the following curve: $x^{2}+(y-b)^{2}=a^{2}$ ($b\geqslant a$) about the $Ox$ axis.

Find the area of the surface formed by rotating the following curve: $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ about the $Ox$ axis.

Find the area of the surface formed by rotating the following curve: $y=a\operatorname {ch}\dfrac {x}{a}$ ($\left|x\right|\leqslant b$):

Find the area of the surface formed by rotating the following curve: $\pm x=a\ln \dfrac {a+\sqrt{a^{2}-y^{2}}}{y}-\sqrt{a^{2}-y^{2}}$ about the $Ox$ axis.

Find the area of the surface formed by rotating the following curve: $x=a(t-\sin t)$, $y=a(1-\cos t)$ ($0\leqslant t\leqslant 2\pi$):

Find the area of the surface formed by rotating the following curve: $x=a\cos^{3}t$, $y=a\sin^{3}t$ about the line $y=x$.

Find the area of the surface formed by rotating the following curve: $r=a(1+\cos \varphi )$ about the polar axis.

Find the area of the surface formed by rotating the following curve: $r^{2}=a^{2}\cos 2\varphi$:

The figure bounded by the parabola $ay=a^{2}-x^{2}$ and the $Ox$ axis is rotated about the $Ox$ axis to form a solid of revolution. Find the ratio of its surface area to the surface area of a sphere of equal volume.

The figure bounded by the line $x=\dfrac {p}{2}$ and the parabola $y^{2}=2px$ is rotated about the line $y=p$. Find the volume and surface area of this solid of revolution.