Methods for Computing Volume

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 $h$.

Find the volume of a paraboloid of revolution whose base is $S$ and whose height equals $H$.

Suppose the area $S=S(x)$ of the cross-section of a solid perpendicular to the $Ox$ axis varies according to the following quadratic law:

where $A$, $B$, and $C$ are constants.

Prove that the volume of this solid equals

where $H=b-a$ (Simpson's formula).

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:

Prove that the volume of the solid of revolution obtained by rotating the area

(where $y(x)$ is a continuous function) about the $Oy$ axis equals

Find the volume enclosed by the surface formed by rotating the following curve: $y=b\left(\dfrac {x}{a}\right)^{\frac{3}{2}}(0\leqslant x\leqslant a)$ about the $Ox$ axis (semicubical parabola).

Find the volume enclosed by the surface formed by rotating the following curve: $y=2x-x^{2}$, $y=0$:

Find the volume enclosed by the surface formed by rotating the following curve: $y=\sin x$, $y=0$ ($0\leqslant x\leqslant \pi$):

Find the volume enclosed by the surface formed by rotating the following curve: $y=b\left(\dfrac {x}{a}\right)^{2}$, $y=b\left|\dfrac {x}{a}\right|$:

Find the volume enclosed by the surface formed by rotating the following curve: $y=e^{-x}$, $y=0$ ($0\leqslant x<+\infty$):

Find the volume enclosed by the surface formed by rotating the following curve: $x^{2}+(y-b)^{2}=a^{2}$ ($0<a\leqslant b$) about the $Ox$ axis.

Find the volume enclosed by the surface formed by rotating the following curve: $x^{2}-xy+y^{2}=a^{2}$ about the $Ox$ axis.

Find the volume enclosed by the surface formed by rotating the following curve: $y=e^{-x}\sqrt{\sin x}$ ($0\leqslant x<+\infty$) about the $Ox$ axis.

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

Find the volume enclosed by the surface formed by rotating the following curve: $x=a\sin^{3}t$, $y=b\cos^{3}t$ ($0\leqslant t\leqslant 2\pi$):

about the $Ox$ axis;

Prove that the volume obtained by rotating the area

(where $\varphi$ and $r$ are polar coordinates) about the polar axis equals

Find the volume obtained by rotating the following area given in polar coordinates: $r=a(1+\cos \varphi )$ ($0\leqslant \varphi \leqslant 2\pi$):

Find the volume obtained by rotating the following area given in polar coordinates: $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$:

about the $Ox$ axis;

about the $Oy$ axis;

Find the volume of the solid of revolution obtained by rotating about the polar axis the area