Volume of frustum cut by an inclined plane at distance h If i have frustum and its top is cut by an inclined plane at angle $\alpha$, such that it makes an ellipse. The height is $h$ (at the axis of obliquely truncated frustum). How can i use triple integral to determine its volume, and coordinates of its geometric center. I will be thankful. I can determine the volume of elliptical cone by using parametric equation however i am confused to obtain parametric equations for right circular cone cut by inclined plane. The radius of the bottom surface is $R$ and top surface is $r$, as shown figure below. Frustum Figure 

 A: The equation of the conical surface can be written in cylindrical coordinates $(\rho, \theta, z)$ as
$$
z={H\over R-r}(R-\rho),
$$
whereas the equation of the plane is
$$
z=(\tan\alpha) \rho\cos\theta+h.
$$
By combining these one can find the equation of the projection of their intersection on the $xy$-plane:
$$
\rho={R-(h/H)(R-r)\over 1+[(\tan\alpha)(R-r)/H]\cos\theta},
$$
which is the polar equation of an ellipse having a focus at $(0,0)$.
In your volume integration, you must integrate along $z$ from $0$ to the value given by the conical surface equation, if $(x,y)$ is outside the ellipse, and from $0$ to the value given by the plane equation, if $(x,y)$ is inside the ellipse.
A: I think the easiest way to get the volume is to find the vertex of the cone from which the frustum is cut, find the volume of the cone between the vertex and the elliptical cut, and subtract that from the volume of the cone between the vertex and the farther face of the frustum.
But if you must have a triple integral, how about setting up a coordinate system better suited to the problem? Consider "conical" coordinates $(\lambda, \theta, z)$, defined in terms of Cartesian coordinates $(x,y,z)$
where the origin is at the vertex of the cone from which the frustum is cut, the $z$ axis is the axis of the cone, 
$\theta=\mathrm{atan2}(y,x)$,
and $\lambda=\frac{\sqrt{x^2+y^2}}{z}.$
In these coordinates, the curved surface of the cone is just a surface with a constant value of $\lambda$.
You can work out the unit of volume integration for this coordinate system; I get
$\lambda z^2\,d\lambda\,d\theta\,dz.$
Find the equation of the inclined plane in this coordinate system in the form $z=f(\lambda, \theta)$,
find $z=z_0$ at the bottom face of the frustum,
and integrate $f(\lambda, \theta)-z_0$ over the values of $\lambda$ and $\theta$ on that face.
