Volume by cross-sectional areas

Find the volume of a hip roof, whose base is a rectangle with side lengths $a$ and $b$, whose ridge has length $c$, and whose height is $h$.

Find the volume of a truncated wedge whose parallel top and bottom faces are rectangles with side lengths $A$, $B$ and $a$, $b$ respectively, and whose height equals $h$.

Find the volume of a truncated cone (frustum) whose top and bottom faces are ellipses with semi-axes $A$, $B$ and $a$, $b$ respectively, and whose height equals $h$.

A solid is the set of points $M(x,y,z)$ such that $0\leqslant z\leqslant 1$, and moreover, if $z$ is rational, $0\leqslant x\leqslant 1$, $0\leqslant y\leqslant 1$; while if $z$ is irrational, $-1\leqslant x\leqslant 0$, $-1\leqslant y\leqslant 0$. Prove that although the corresponding integral is

the volume of this solid does not exist.

Find the volume enclosed by the following surfaces:

Find the volume enclosed by the following surfaces:

Find the volume enclosed by the following surfaces:

Find the volume enclosed by the following surfaces:

Find the volume enclosed by the following surfaces:

Find the volume enclosed by the following surfaces:

Find the volume enclosed by the following surfaces:

Find the volume enclosed by the following surfaces:

Find the volume enclosed by the following surface:

Liquid spills over the edge of a vessel in the shape of a cylinder with radius $a$ and height $h$. Prove that at the moment when half of the bottom becomes exposed, the volume of the remaining liquid equals $\frac{2}{3}a^{2}h$. (Archimedes.)

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.

Find the volume of the solid bounded by the surfaces.

Find the volume of the solid bounded by the surfaces.

Find the volume of the solid bounded by the surfaces.

The base of a cone is a plane figure of area $S$, the vertex of the cone is at distance $H$ from the plane of the base. Find the volume of the cone.

The radius of a right circular cylinder equals $r$. A plane, making an angle $\alpha$ with the axis of the cylinder, intersects its lateral surface and lower base, cutting off from this base a segment with central angle $2 \varphi$. Find the volume of the part of the cylinder bounded by this segment, the cutting plane, and the lateral surface of the cylinder.

The opening angle of a right circular cone equals $2 \alpha (2 \alpha < \pi / 2)$. A plane perpendicular to a generator of the cone intersects it at a point at distance $a$ from the vertex. Find the volume of the part of the cone cut off.

Through the center of the base of a right circular cone with radius $r$ and height $h$ a plane parallel to a generator of the cone is drawn. Find the ratio of the volumes of the parts into which the cone is divided.

Find the volume of the solid bounded by the surfaces:

The axes of an ellipse have lengths $2 a$ and $2 b$. On every chord of the ellipse parallel to the axis of length $2 a$, a square is constructed as on a diagonal, in the plane perpendicular to the other axis of the ellipse. Find the volume of the resulting solid.

Two faces of a hexahedron are rectangles, the sides of one of them of lengths $a_{1}$ and $b_{1}$ being parallel to the sides of the other of lengths $a_{2}$ and $b_{2}$ respectively. The distance between the planes of these rectangles equals $h$. Find the volume of the hexahedron.

The volume of the triangular prism $A B C A_{1} B_{1} C_{1}$ equals $V$. Consider all triangles lying in planes parallel to the bases, with vertices on the diagonals $A B_{1}, B C_{1}$ and $C A_{1}$ of the lateral faces of the prism. Find the volume of the solid formed by these triangles.

A solid, having a certain volume, is located between the planes $z = 0$ and $z = h$. It is known that the area of the cross-section of the solid by the plane $z =$ const is a function of the form $S(z) = a z^{2}+b z+c, 0 \leqslant z \leqslant h$. Prove that the volume of the solid equals

The axes of two circular cylinders, each of which has radius $r$, intersect at an angle $\alpha$. Find the volume of the solid bounded by these cylinders.

On a circle $x^{2}+y^{2} \leqslant r^{2}$, as on a common base, two oblique cylinders are constructed. Their axes lie in the plane $O y z$, and each makes an angle $\alpha$ with the axis $O z$. Find the volume of the solid bounded by the given circle and these cylinders.

The axes of three cylinders (each of radius $r$) are mutually perpendicular and intersect at one point. Find the volume of the solid bounded by these three cylinders.

On the lateral surface of the cylinder $x^{2}+y^{2} \leqslant R^{2}, 0 \leqslant z \leqslant H$ lies a helix $x = R \cos 2 \pi \alpha , y = R \sin 2 \pi \alpha , z = H \alpha$, $0 \leqslant \alpha \leqslant 1$. A helical surface is formed by the perpendiculars dropped from the points of the helix onto the axis $O z$. Find the volume of the lower part of the cylinder bounded by this surface and the rectangle $0 \leqslant x \leqslant R, 0 \leqslant z \leqslant H$ in the plane $O x z$.

On the lateral surface of the cone

lies a conical helix

A conical helical surface is formed by the perpendiculars dropped from the points of a helix onto the axis $O z$. To it is added a triangle with vertices $(0 ; 0 ; 0),(0 ; 0 ; H),(R ; 0 ; 0)$. Find the ratio of the volumes of the parts into which the resulting surface divides the cone.