Test review

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