1) A plate is bounded by the curves y = -x^2, y = x^2, and x = 1. It has density d(x,y)=x. Make a sketch and find its center of mass.
2) A solid occupies a half-cylinder W: x^2 + y^2 is less than or equal to 4, y is greater than or equal to 0, z is greater than or equal to -3 and less than or equal to 3. It has constant density d(x,y,z) = 4. Find its mass and center of mass (x-bar, y-bar, z-bar). Use cylindrical coordinates and a picture.
Density and Center of Mass are investigated. The solution is detailed and well presented.