Test review

MATH 3433 - CALCULUS III ONLINE 5) [15 Points Each] Setup (in an appropriate coordinate system and JUSTIFY Exam 4 your choice of the coordinate system) the integrals necessary to accomplish the Name: following tasks. You are not evaluating these integrals. Date: . Problem Set 1: a 3D object has density function p(x, y, z) = 2xyer2+ +z Use blank paper to complete this test - make sure problems are all labeled clearly and It occupies the first octant below a cone whose walls are 60 above the ry- submitted in order. You must show all your work in a clear format to receive full credit plane and inside a sphere of radius 3. Set up the integrals that will calculate on each problem. Remember to simplify all answers. the following. - Mass (1) [5 Points] Explain (using 1 - 3 sentences and a picture) why there is an integration - All three first moments factor involved in some coordinate systems but not others when evaluating a double - Center of mass integral in dA or a triple integral in dV. It might help to show how dA / dV relates Problem Set 2: a 2D object has density function p(x, y) = x y + y3. It to the appropriate differentials in a given coordinate system. Note that this is not occupies the space inside the isosceles triangle connecting the points (-3, 0), a calculation of what the integration factors are, simply an explanation of why they (3, 0), and (0, 9). Set up the integrals that will calculate the following. exist. - Mass - All three sccond moments (2) [10 Points Each] Sketch and describe the domain region illustrated by the fol- - Both radii of gyration lowing integrals. . Problem Set 3: a joint density function for probability of three events is given by the following equation: f (x, y, z ) = 12x2yz [0, 1] x [0, 1] x [0, 1] otherwise Answer the following questions - set up any integrals necessary to do so. 3 [ - f (x, y ) di dy - Explain (and set up any integrals necessary) how to verify this function is in fact a probability density function Probability that a chosen point (x, y, z) is such that 2x + y