Name:Lab Section:MATH 215 – Winter 2005SECOND EXAMShow your work in this booklet.Please do NOT submit loose sheets of paper–They won’t be gradedProblem Points Score1 202 153 204 155 156 15TOTAL 100Some useful trigonometric identities:sin2θ + cos2θ = 1 cos 2θ = cos2θ − sin2θ sin 2θ = 2 sin θ cos θsin2θ =1 − cos 2θ2cos2θ =1 + cos 2θ2Spherical coordinates:x = ρ cos(θ) sin(φ) y = ρ sin(θ) sin(φ) z = ρ cos(φ)1Problem 1. (5+5+10=20 points) Throughout this problem f is the functionf(x, y) = xye−x−2y.(a) Find all critical points of f.(b) A calculation shows thatfxx= e−x−2y¡−2y + xy¢, fxy= e−x−2y¡1 − x − 2y + 2xy¢, fyy= e−x−2y¡−4x + 4xy¢.Using these formulas, classify the critical points of f. You do not have to compute the second partialsyourself.2(c) Find the absolute maximum of the function f on the triangle with vertices at (0, 0), (1, 0) and(0, 1/2). Hint: There are not any critical points in the interior of the triangle.3Problem 2. (15 points) Suppose that the temperature in Celsius at the point (x, y, z) of theellipsoid 4x2+ y2+ 4z2= 6 isT (x, y, z) = 2x + 1y − 4z + 600.Find the hottest and coldest points on the ellipsoid, and the highest and lowest values of thetemperature.4Problem 3. (10+10=20 points) This problem consists of two unrelated questions on doubleintegrals.(a) Evaluate the double integralRRDxy2dA if D is the region in the half plane x ≥ 0 bounded bythe curves x = 1 − y2and x2+ y2= 1.0.2100 0.60.5-0.5-110.80.4(b) Change the order of integration in the iterated integralZ20Z8x2x3f(x, y) dy dx. (Do not attemptto evaluate the resulting integral.)5Problem 4. (8+7=15 points) This problem is about finding the the Cartesian coordinates (x, y)of the center of mass of a homogeneous lamina enclosed by the cardioid r = 1 − cos(θ) (see figure).100.5-0.5-10-0.5-1-1.5-2(a) Find the area of the lamina.(b) Give the value of y, and express x in terms of an iterated double integral where the limits ofintegration are explicitly determined. Please circle your answer.6Problem 5. (15 points) In this problem we use the notation:I =ZZZEf(x, y, z) dV,where E is the region in the first octant bounded above by the plane x + 2y + 3z = 6.Express I as an explicit iterated triple integral of the formR R Rdz dy dx. Circle your answer. Donot attempt to evaluate the integral. Make sure you specify all the limits of integration.7Problem 6. (5+10= 15 points) Let E be the region in the first octant in space, inside the spherex2+ y2+ z2= R2and above the cone 3z2= x2+ y2.(a) Find the equation of the cone in spherical coordinates.(b) Find the volume of the