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MATH 023 Calculus III Fall 2020Modified Old Examination IVExamination IV will take place Wednesday, November 4, during yourZoom class meeting.The relevant sections from the book are 12.1–12.6,13.1–13.4, 14.1–14.8, 15.1–15.5, and 16.1–16.5, although the emphasis willbe on 15.1–15.5 and 16.1–16.5 (please see the syllabus, the schedule, andthe class notes for more information). Warning: The upcoming examina-tion may be different.1.A lamina (i.e., a thin plate) occupies the unit diskDcentered at the origin (0,0)inR2. The density at each point (x, y) is proportional to the distance between thepoint and the liney=-3. Find the center of mass.Hint.Use symmetry to save time, but only where appropriate.2.Sis the part of surfacez= 16 +y2-x2that lies above the triangle with vertices(0,0), (4,0), and (1,2). Set up, butDO NOT EVALUATE, an iterated integral(or iterated integrals) giving the area ofS.3.Determine whether or not the vector fieldF(x, y, z) =hyz+y2, xz+y, xy+eziisconservative, and if so, find a functionfsuch that∇f=F.4.Suppose that:•P(x, y) andQ(x, y) are functions that are continuously differentiable at everypoint inR2except(0,0) and (0,3), and that satisfy∂P∂y=∂Q∂x;•Ais the circle of radius 5 and center (0,0), directed counterclockwise;•Bis the circle of radius 1 and center (0,0), directed counterclockwise;•Cis the circle of radius 1 and center (0,3), directed counterclockwise; and•ZAP dx+Q dy= 9 andZBP dx+Q dy= 15.EvaluateZCP dx+Q dy. As always, remember to give complete reasons.Z2Z-32y+42ypAnswers (not solutions): 1.(0,112).2.14x2+ 4y2+ 1dx dy.3. Fis not conservative.4.-6.1