**Problem #1: **A lamina in the shape of the region bounded byand the line* x* = 0. The areas density varies as the distance from the *x*-axis. Find the center of mass. Please provide an accurate sketch of the bounded region.

**Solution: **We first compute the intersection points:

The only real solution is approximately .

Then, we haveOn the other hand, by similar integration:

andThis implies that the center of mass is

**Problem #2: **Find the area of the surface of the portion of the conebetween the cylinderand the plane.

**Solution: **We have that the projection of the region we need to calculate is enclosed in the *x*–*y* plane byand . We need to intersect these curves:

which means that. Therefore, we have the following parametric representation of the surface:

where. We have that

and the normal vector is computed as

Computing the norm:

This means that the area is

**Problem #3: **Use polar coordinates to evaluate, where *R* is the region bounded by the circlesand.

**Solution: **We use polar coordinates