1. Let S be a sphere of radius a centered at the origin. (a) Evaluate the following integral (b) Now take the limit as a - co. Use this result to determine the value of 2. The Jacobian for the change of variables x = g(u, v), y = h(u, v) is given by a(x, y) ax ay ay Or J(u,") = (u, v) du du du du (a) Calculate J(r,8) if a = reos(0) and y = rain(@) (b) The Jacobian is used to implement a change of variables when integrating. If the region of integration is D and is given by a and y, then the new region D' will be given by a and a according to a = g(u, v) and y = h(u, v). The integration would change as follows Now, use the change of variable z = ar cos, y = brain 0 to evaluate where S is the quarter ellipse - 1 1, 2 20, 120 O 83 N W 52OF3. Use question 2 to answer the following question. Find the volume of the region bounded by 2 + 3y' - z and y' + 2 = 4. 4. Extension to higher dimensions. Let H be the four-dimensional region given and evaluate the following: myzwe da dy de du where H is the four-dimensional hyperbox defined by