Past Exam Problems in IntegralsProf. Qiao ZhangCourse 110.202November 15, 2004The following is a list of the problems concerning integrals that appearedin the midterm and final exams of Calc III (110.202) within the last sev-eral years. You may use them to check your understanding of the relevantmaterial. Some other exam problems may b e found athttp://reserves.library.jhu.edu/access/reserves/findit/exams/110/110202.phpNote: These problems do not imply, in any sense, my taste or prefer-ence for our own exam. Some of the problems here may be more (or less)challenging than what will appear in our exam.1. Evaluate the double integralZZDx2y2dx dyover the triangle D with vertices (0, 0), (1, 0), (1, 2).2. The shape of a platform is given byx2+ y2≤ (2 − z)2, 0 ≤ z ≤ 1.(a) Describe this shape in cylindrical coordinates.(b) Find the volume of this platform.3. Find the volume of the solid enclosed by the two paraboloids z =2(x2+ y2) and z = 1 + x2+ y2.14. Find a parametrization for the surface defined by the intersection ofthe plane x + y + z = 1 with the cylinder x2+ y2= 1. Use thatparametrization to calculate the area of the surface.5. Suppose that a particle follows the path r(t) = 2 cos(2t)i + 2 sin(2t)j +3tk. Then find the total length of the path travelled by the particlefrom t = 0 to t = π/4.6. Set up a double integral in polar coordinates to find the volume of thesolid which is bounded below by the paraboloid z = x2+ y2and aboveby the plane z = 2y. DO NOT EVALUATE!7. Evaluate the integralZ10Z1ycos12πx2dx dy.8. Find the volume of the region under the surface z = x2+ 2y and overthe region in the first quadrant between the line segment connecting(0, 2) and (2, 0) and the curve y = 4 −