Calculus

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

 

 

 

 

 

 

Volume =

 

  1. 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

=,

==

 

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:

=