Sample Solved Calculus Problems

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 xy 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