Problem 2.1 Area of triangle

Consider two points in 3d, A and B. Show that the area of the triangle spanned by A, B and the origin in terms of the components of these vectors is given by area (A (0 A B) ) = | AXBI .A paraboloid emerges by rotating a parabola about its symmetry axis, see Figure 2.21. Con- sider the homogeneous paraboloid consisting Vab -+ of (x, y, z) E R' satisfying x2 + yz s az b for 0 0. Determine its volume. Use symmetry arguments to con- clude that the center of mass lies on the z-axis. Figure 2.21: Paraboloid. Compute the location of the center of mass.m2 1 = R+ M m1 2) = R M