1. The level curves for the solid, projected on an x-y plane :

Volume =

*I*To 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

=,

==

3.a.

Note that cubic roots, constrained to R are unique. We do not need two seta of parametrisation.

Consider:

We can write the integral as

Limits of (u,v) : (-1/a ,1/a),

F(x) is unbounded.

I=

2..

It is straight forward to integrate f(x) given the limits on the plane on y over the height of cylinder

Limits:

Volume =

4. The given integral can be represented as, by cros-substitution of variables as:

=