I need to give a sage math code and show all steps and the work out in detail for each part , for the code don't give only results , give the whole code with the output

4 Taking it up a notch (#integration, #computationaltools) Just like we went from a single to a double integral, we can add another dimension to create a triple integral. One (hard-to-visualize) interpretation of a triple integral is the hyper-volume of a 4-dimensional region. We can interpret triple integrals in other ways depending on the context of the problem. (a) Give a qualitative interpretation of [f] p [ dV for each of the following descriptions of f. 1. f(x,y,2) = 2 forall points in a 3-dimensional region R. Hint: Recall the meaning of the double integral of f(x,y) = 2. 2. f(z,y, z) is the charge density at point (i, y, 2) of an object occupying a region R. 3. f(x,y, z) is the joint probability density function for three independent random variables. (b) Consider three non-negative random variables, U, V, W. Let f(u,v,w) = k- u*-v* - w inside the following region R and 0 elsewhere. The region R is defined as the region under the plane w = 5 u v and in the first octant (u 2> 0, v > 0, w > 0). What value must k take such that f(u, v, w) satisfies all the conditions to be a probability density function? (c) Compute the probability P(U > W). (d) Find the expected values of U, V, W, denoting them as U, V, W respectively