1. The level curves for the solid, projected on an x-y plane :
- ITo help us integrate, we use the cylindrical co-ordinate,
The three coordinates of a point P are defined as:
- The radial distance r is the Euclidean distance from the z axis to the point P.
- The azimuth is the angle between the reference direction on the chosen plane and the line from the origin to the projection of P on the plane.
- The height x is the signed distance from the chosen plane to the point P.
Rotational Inertia: I=
Now radius of each slice varies along h, r= a/h*x.
We switch the order of integration, determining the limits
Note that cubic roots, constrained to R are unique. We do not need two seta of parametrisation.
We can write the integral as
Limits of (u,v) : (-1/a ,1/a),
F(x) is unbounded.
It is straight forward to integrate f(x) given the limits on the plane on y over the height of cylinder
4. The given integral can be represented as, by cros-substitution of variables as: