Find the mass and center of mass of the tetrahedron bounded by the planes x = 0, y = 0, z = 0, 3x + y + 2z = 6, if the density function is given by p(x, y, z) : = 2.

Find Mx, My, and (x, y) for the laminas of uniform density p bounded by the graphs of the equations. x = -Y, x = 5y - y Mx = My =

The motion of a solid object can be analyzed by thinking of the mass as concentrated at a single point, the center of mass. If the object has density p(x, y, z) at the point (x, y, z) and occupies a region W, then the coordinates (x, y, z) of the center of mass are given by 1 = m η χαραν xp dV y = yp dV = z zpdV, m m where m = SwpdV is the total mass of the body. Consider a solid is bounded below by the square 2 = 0, 0 ≤ x ≤3,0≤ y ≤2 and above by the surface z = x + y + 5. Let the density of the solid be 1 g/cm³, with x, y, z measured in cm. Find each of the following: The mass of the solid = for the solid = y for the solid = z for the solid = (For each, include units.)

Use Green's theorem to compute the circulation of F(x, y) = (-y, x²) around the perimeter of the square [0, 2] × [0, 2].