Computing Volumes

Find the volume of the solid obtained by revolving the hyperbola $xy=a^{2}$ about the axis $Ox$ for $a<x<\infty$.

Find the volume of the solid obtained by revolving the sine curve $y=\sin x$ about the axis $Ox$ for $0<x<\pi$.

Find the volume of the solid obtained by revolving the cissoid $(2a-x)y^{2}=x^{3}$ about $Ox$ for $0<x<b<2a$.

Find the volume of the solid obtained by revolving the lemniscate $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$ about the axis $Ox$.

Find the volume of the solid obtained by revolving the cardioid $r=a(1+\cos \varphi )$ about the polar axis.

Find the volume of the solid obtained by revolving the cycloid

about the axis $Ox$, for $0<x<2\pi a$.

Find the volume of the solid obtained by revolving the parabola $y^{2}=2px$ about the axis $Ox$ for $0<x<a$.

Find the volume of the solid obtained by revolving the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ about the axis $Ox$.

Find the volume of the solid obtained by revolving the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ about the axis $Ox$ for $a<x<m$.

Find the volume of the solid obtained by revolving the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ about the axis $Oy$ for $0<y<h$.

Find the volume of the solid obtained by revolving the cycloid

about the line $x=\pi a$.

The cycloid

is revolved about the intersecting line $y=ka$, where $0<k<2$. Determine the volumes of the resulting solids of revolution.

Find the volume obtained by revolving one loop of the lemniscate $r^{2}=a^{2}\sin 2\varphi$ about the polar axis.

The curve $y=\frac{4a^{3}}{x^{2}+4a^{2}}$ is revolved about its asymptote. Find the volume of the resulting infinite spindle.

The curve $r^{3}=a^{3}\cos 3\varphi$ is revolved about the polar axis. Determine the volume of the solid obtained by revolving the loop lying in the third quadrant.

An arc of the parabola $\sqrt{x/a}+\sqrt{y/b}=1$ is revolved about the line $\frac{x}{a}+\frac{y}{b}=1$. Prove that the volume of the resulting solid of revolution is $V=\frac{8}{15}S_{M}\cdot H$, where $S_{M}$ is the area of the largest circular cross-section, and $H$ is the length of this spindle-shaped solid.

A segment of a circle of radius $R$, corresponding to a central angle $2\alpha$, is revolved about its chord. Determine the volume of the resulting solid.

The limaçon of Pascal $r=a+b\cos \varphi$ is revolved about the polar axis. Determine the volumes:

Find the volume of the solid bounded by the surface obtained by revolving an arc of the parabola $y^{2}=4ax$ about the line $y=2x$.

A cube with edge $a$ is revolved about its space diagonal. Determine the volume of the solid obtained by revolving one face.

A cube with edge $a$ is revolved about the diagonal of one of its faces. Determine the volume of the solid obtained by revolving the opposite face.

The curve $x^{4}+y^{4}=2axy^{2}$ is revolved about the axis $Oy$. Determine the volume of the resulting solid.

The same problem as 3419, but the curve is revolved about the axis $Ox$.

Problem 3419: the curve $x^{4}+y^{4}=2axy^{2}$.

Find the volume of the solid bounded by the surfaces $z^{2}+y^{2}=1$, $x^{2}+z^{2}=1$, by considering horizontal cross-sections.

Find the volume of the solid bounded by the surfaces $x^{2}+y^{2}=ax$, $x-z=0$, $x+z=0$, by considering cross-sections perpendicular to $Ox$.

Find the volume of the solid bounded by the surfaces $x^{2}+4y^{2}=8z$, $x^{2}+4y^{2}=1$, $z=0$.

Find the volume of the solid bounded by the surfaces $y^{2}=2p(a-x)$, $x-z=0$, $x-2z=0$.

Find the volume of the solid bounded by the surfaces $z^{2}=(a-x-y)a$; $x=0$, $y=0$, $z=0$.

Find the volume of the solid bounded by the surfaces $z^{2}=b(a-x)$, $x^{2}+y^{2}=ax$.

Find the volume of the solid bounded by the surfaces $z=0$, $by=x(a-z)$, $by=-x(a-z)$, $x=b$.

Find the volume of the solid bounded by the surfaces $y^{2}+z^{2}=a^{2}\ch^{2}\frac{x}{a}$, $-b<x<b$.

Water is poured into a right circular cylinder (a glass) of radius $r$. The axis is tilted at an angle $\alpha$ to the horizontal. Part of the bottom — a segment with central angle $2\varphi$ — is covered by water. Find the volume of the water.

A right circular cone is cut into two parts by a plane passing through the center of the base and parallel to a generator. Find the volumes of the parts of the cone, given that the cross-sections of the cone by planes parallel to the generator are parabolic segments.

A quarter of a right circular cylinder is cut by an inclined plane passing through a radius of the upper base and the endpoint of the radius of the lower base perpendicular to the first radius. Through the axis of the cylinder a plane is drawn at an angle $\varphi$ to the aforementioned radius of the upper base. Find the volumes of the four parts of the quarter-cylinder.

Find the volume bounded by the surface of the conoid $a^{2}l^{2}y^{2}=b^{2}x^{2}(a^{2}-z^{2})$ and the plane $x=l$.

Three mutually perpendicular lines passing through a single point are the axes of three circular cylinders of the same radius $r$. Determine the volume of the solid lying inside all of them.

The axes of two circular cylinders of radius $r$ intersect at an angle $2\alpha$. Determine the volume of the solid bounded by these cylinders.