MATH 228: Calculus III SAMPLE MIDTERM II Spring 2014 NAME : NOTE : There are 5 problems on this midterm (total of 6 pages). Use of calculators will NOT be permitted. In order to receive full credit for any problem, you must show work leading to your answer. You have 50 minutes to complete this test. Problem Possible points 1 20 2 20 3 20 4 20 5 20 Total 100 Score MATH 228 – MIDTERM II Page 2 Problem 1. (20pts) (a) (10pts) Sketch the region in the xy-plane whose area is given by the iterated integral Z 6 Z 2y dx dy 0 y 2 /3 (b) (10pts) Evaluate the iterated integral. MATH 228 – MIDTERM II Page 3 Problem 2. (20pts) (a) (10pts) Change the Cartesian integral into an equivalent polar integral. Z 2Z x y dy dx 0 0 (b) (10pts) Evaluate the polar integral. MATH 228 – MIDTERM II Page 4 Problem 3. (20pts) Setup and evaluate an intreated integral for the volume of the tetrahedron in the first octant bounded by the coordinate planes and the plane passing through (1, 0, 0), (0, 2, 0) and (0, 0, 3). MATH 228 – MIDTERM II Page 5 Problem 4. (20pts) E is the solid right circular cylinder whose base is the circle r = 2 sin θ in the xy-plane and whose top lies in the the plane z = 4−y. Setup an iterated integral for evaluating (BUT DO NOT EVALUATE) Z Z Z z dV E MATH 228 – MIDTERM II Page 6 Problem 5. (20pts) Let E be the smaller cap cut from a solid ball of radius 2cm by a plane a distance 1cm from the center of the sphere. Compute the volume of E.