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
Best Answer
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.