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# Sample Solved Calculus Problems Problem #1: A lamina in the shape of the region bounded by and 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: and This implies that the center of mass is Problem #2: Find the area of the surface of the portion of the cone between the cylinder and the plane .

Solution: We have that the projection of the region we need to calculate is enclosed in the xy plane by and . 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 circles and .

Solution: We use polar coordinates Hence, we get 