Computing Volumes

Find the volume bounded by the surfaces $x+y+z=6$, $x=0$, $z=0$, $x+2y=4$.

Find the volume bounded by the surfaces $x-y+z=6$, $x+y=2$, $x=y$, $y=0$, $z=0$.

Find the volume bounded by the surfaces $x+y+z=6$, $x=y$, $y=0$, $x=3$, $z=0$.

Find the volume bounded by the surfaces $z=a+x$, $z=-x-a$, $x^{2}+y^{2}=a^{2}$.

Find the volume bounded by the surfaces $z^{2}=xy$, $x=a$, $y=b$, $z=0$.

Find the volume bounded by the surfaces $x+y+z=a$, $3x+y=a$, $3x+2y=2a$, $y=0$, $z=0$.

Find the volume bounded by the surfaces $x+y+z=a$, $x^{2}+y^{2}=b^{2}$, $z=0$; $a>b\sqrt{2}$.

Find the volume bounded by the surfaces $\dfrac {x^{2}}{a^{2}}+\dfrac {z^{2}}{c^{2}}=1$, $y=\dfrac {b}{a}x$, $y=0$, $z=0$; $x>0$.

Find the volume bounded by the surface $x^{2}z^{2}+a^{2}y^{2}=c^{2}x^{2}$, $0<x<a$.

Find the volume bounded by the surfaces $y^{2}+z^{2}=x$, $x=y$; $z>0$.

Find the volume bounded by the surfaces $z=x^{2}+y^{2}$, $y=x^{2}$; $0<y<1$, $z=0$.

Find the volume bounded by the surfaces $z=\cos x \cos y$; $\left|x+y\right|<\dfrac {\pi }{2}$, $z=0$.

Find the volume bounded by the surfaces $z=\sin (x^{2}+y^{2})$, $z=0$; $n\pi <x^{2}+y^{2}<(n+1)\pi$.

Find the volume bounded by the surfaces $x^{2}+y^{2}=cz$, $x^{2}+y^{2}=ax$, $z=0$.

Find the volume bounded by the surfaces $x^{2}+y^{2}=2ax$, $z=\alpha z$, $z=\beta x$; $\alpha <\beta$.

Find the volume bounded by the surfaces $x^{2}+y^{2}=\alpha z^{2}$, $x^{2}+y^{2}=ax$; $z>0$.

Find the volume bounded by the surfaces $x^{2}+y^{2}=az$, $x+z=2a$.

Find the volume bounded by the surfaces $cz=xy$, $x^{2}+y^{2}=ax$, $z=0$; $y>0$.

Find the volume bounded by the surfaces $z=x^{2}+y^{2}$, $z=x+y$.

Prove that the volume bounded by the plane $z=mx+ny+p$ and the paraboloid $z=x^{2}+y^{2}$ equals half the area of the circle $x^{2}+y^{2}=mx+cy+p$, multiplied by the difference of the quantities $x^{2}+y^{2}$ and $mx+cy+p$, taken at the center of that circle.

Prove that the volume of the segment cut off by a plane from an elliptic paraboloid equals the area of the base of the segment, multiplied by half the height of the segment.

A part is cut from a circular cone with base radius $R$ and height $H$ by a plane parallel to the axis of the cone. Determine the volume of the cut-off part, if the segment at the base corresponds to a central angle $2\alpha$.

A right prism with a square base (of side $a$) and height $H$ is intersected by a circular cone in such a way that its trace on one base is the circle inscribed in the base, and on the other base is the circle circumscribed about it. What fraction of the volume of the prism lies outside the cone?

A right hexagonal prism (a nut) with face width $a$ is intersected by a coaxial conical surface whose generator makes an angle $\alpha =60^{\circ }$ with the axis. Determine the volume between the lateral surface of the prism, its base, and the cone, if the cone intersects the base along the circle inscribed in the base. In the following problems, as in some of the preceding ones, it is useful to introduce polar coordinates.

Find the volume bounded by the surfaces $x^{2}+y^{2}=cz$, $x^{4}+y^{4}=a^{2}(x^{2}+y^{2})$, $z=0$.

Find the volume bounded by the surfaces $z^{2}=2xy$, $(x^{2}+y^{2})^{2}=2a^{2}xy$, $z=0$; $x>0$, $y>0$.

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

Find the volume bounded by the surfaces $x^{2}+y^{2}+z^{2}=a^{2}$, $x^{2}+y^{2}>\left|ax\right|$. (Viviani's problem.)

Find the volume bounded by the surfaces $z=x^{2}+y^{2}$, $z^{2}=xy$; $x>0$, $y>0$.

Find the volume bounded by the surfaces $z(x+y)=ax+by$, $z=0$; $1<x^{2}+y^{2}<4$, $x>0$, $y>0$, $a>0$, $b>0$.

Prove that the volume of the body bounded by the surfaces:

where $\varphi (x)$ is any positive integrable function, $a>0$ and $b>0$, equals $\dfrac {1}{2}\pi c^{2}(a+b)$.

Prove that the volume bounded by the surfaces $z=0$ and $z=e^{-x^{2}-y^{2}}$ equals $\pi$.

By a linear change of variables, find the volume enclosed between the surfaces $z=0$ and $z=Ae^{-(ax^{2}+bxy+cy^{2})}$, where $a>0$, $b^{2}-4ac<0$.

Find the volume bounded by the surfaces $\dfrac {x^{2}}{p}+\dfrac {y^{2}}{q}=2z$, $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$, $z=0$.

Find the volume bounded by the surfaces $cz=xy$, $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$, $z=0$; $x>0$, $y>0$.

Find the volume bounded by the surface $\left(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}\right)^{2}+\dfrac {z^{2}}{c^{2}}=1$.

Find the volume bounded by the surface $\left(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}\right)^{k}+\dfrac {z}{c}=1$; $z>0$; $k>0$.

Find the volume bounded by the surfaces $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\dfrac {z^{2}}{c^{2}}$, $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\dfrac {x}{h}$; $z>0$.

Find the volume bounded by the surfaces $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\dfrac {z}{c}$, $\dfrac {x^{4}}{a^{4}}+\dfrac {y^{4}}{b^{4}}=\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}$, $z=0$.

Find the volume bounded by the surfaces $z^{2}=2xy$, $\left(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}\right)^{2}=\dfrac {2xy}{c^{2}}$; $x>0$, $y>0$, $z>0$.

Find the volume bounded by the surfaces $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\dfrac {z}{c}$, $\dfrac {x}{a}+\dfrac {z}{c}=2$.

Find the volume bounded by the surface $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{2}+\dfrac {z^{2}}{c^{2}}=1$; $x>0$, $y>0$, $z>0$.

Find the volume bounded by the surface $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{4}+\dfrac {z^{2}}{c^{2}}=1$; $x>0$, $y>0$, $z>0$.

Find the volume bounded by the surfaces $z=x\sqrt{x}+y\sqrt{y}$, $x+y=1$; $x>0$, $y>0$, $z>0$.

Find the volume bounded by the surfaces $z^{2}=xy$, $x+y=1$; $z>0$.

Find the volume bounded by the surfaces $cz=xy$, $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{4}=\dfrac {xy}{ab}$; $x>0$, $y>0$, $z>0$.

Find the volume bounded by the surfaces $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{2}+\dfrac {z^{2}}{c^{2}}=1$, $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{2}=\dfrac {x}{a}$; $y>0$, $z>0$.

Find the volume bounded by the surface

Find the volume bounded by the surface $\left(\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}\right)^{8}+\left(\dfrac {z}{c}\right)^{2}=1$; $x>0$, $y>0$, $z>0$.

Find the volume bounded by the surface $\left(\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}\right)^{4}+\left(\dfrac {z}{c}\right)^{2}=1$; $x>0$, $y>0$, $z>0$.

Find the volume bounded by the surfaces $z=ye^{-\frac{xy}{a^{2}}}$, $xy=a^{2}$, $xy=2a^{2}$, $y=m$, $y=n$, $z=0$.

Find the volume bounded by the surfaces $cz=xy$, $y^{2}=mx$, $y^{2}=nx$, $x=\alpha y$, $x=\beta y$, $z=0$.

Find the volume bounded by the surfaces $z^{2}=xy$, $xy=a^{2}$, $xy=4a^{2}$, $x=2y$, $x=3y$, $z=0$.

Find the volume bounded by the surfaces $cz=xy$, $y^{2}=2x$, $y^{2}=3x$, $x^{2}=y$, $x^{2}=2y$, $z=0$.

Find the volume bounded by the surfaces $z^{2}=xy$, $xy=1$, $xy=4$, $y^{2}=x$, $y^{2}=3x$, $z=0$.

Find the volume bounded by the surfaces $z=x^{2}y$, $y^{2}=a^{2}-2ax$, $y^{2}=m^{2}+2mx$, $y=0$, $z=0$.