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 have
On the other hand, by similar integration:
This 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
Hence, we get