Advanced Calculus fi..

Chapter 4 Integral Calculus of Functions of Several Variablesc) z=xZy, 05x51, x+liyix+2d) z=,/m-, x2-y2,~, O(x< 13. For each of the following choice of R, represent JJ f (x, y) dA over R as an iteratedintegral of both forms (4.33) and (4.34):a) 15x52, 1-xiyll+xb) y2+x(x- 1)(04. Evaluate jjjR f (x, y, z) d V for the following choices of f and R:a) f (x, y, z) = J-, R the cube of vertices (O,0, O), (LO, O), (0,1, O), (0,0, I),(1, 1,O). (l,O, I), (0, 1, l), (1. 1, 1).b) f (x, y, z) = x2 + z2, R the pyramid of vertices ( f 1, f 1,O) and (0,0, 1).5. For each of the following iterated integrals, find the region R and write the integral inthe other form (interchanging the order of integration):a) j,',2&-X f(x,y)dydxI!1 I+*d) So j,-, f(x,y)dydx6. Express the following in terms of multiple integrals and reduce to iterated integrals, butdo not evaluate:a) the mass of a sphere whose density is proportional to the distance from one diametralplane;b) the coordinates of the center of mass of the sphere of part (a);C) the moment of inertia about the x-axis of the solid filling the region 0 5 z 5 1 -x2 - y2, 0 5 x 5 1,O 5 y 5 1 - x and having density proportional to xy.7. The moment of inertia of a solid about an arbitrary line L is defined asJwhere f is density and d is the distance from a general point (x, y, z) of the solid to theline L. Prove the Parallel Axis theorem:where is a line parallel to L through the center of mass, M is the mass, and h is thedistance between L and Z.,(~int: Take to be the z axis.)8. Let L be a line through the origin 0 with direction cosines 1, m, n. Prove thatwhereIxy = /fl xyf (x, y, z)dx dy dz. I,, = /fl yzf . . . .R -- RThe new integrals are called products of inertia. The locusIxx2 + lyy2 + zzz2 - 2(Ixyxy + I,,yz + Izxzx) = 1is an ellipsoid called the ellipsoid of inertia.