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Review sheet for test 4 (v1).pdf - Math 241 Review sheet...

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Math 241Review sheet for Test 4November 12, 20221. LetCbe the curve in three-dimensional space parametrized byr(t) =2 cost,2 sint, tforπtπ.(a) Find the mass of a thin wire in the shape ofC, if the density function isρ(x, y, z) =x+z+ 10.(b) Suppose that a particle moves alongCstarting atr(π) = (2,0,π) and ending atr(π) =(2,0, π). Find the work done on the particle by the forceF(x, y, z) =yixj.2. LetF(x, y) =3y22x,6xy, and letCbe the linear path from (1,0) to (0,1).(a) EvaluateZCF·drdirectly. (Do not use potential functions.)(b) Show thatFis conservative by finding a potential function for it. Use a systematic method to findyour potential function.(c) EvaluateZCF·drusing your potential function from the previous part.3. LetCbe the triangle in thexy-plane with vertices at (0,0), (1,0), and (0,1), oriented counter-clockwise.(a) EvaluateIC2y2dx+ 2x dydirectly as a line integral.(b) Evaluate the line integral in the previous part by using Green’s Theorem and then evaluating adouble integral.4. LetSbe the surface parameterized byr(u, v) =vcosu, v, vsinufor 0u2πand 0v1.(a) Mark the picture ofSbelow.(b) Evaluate the surface integralZZSy dS.5. LetZbe the part of the graph ofz=g(x, y) = 2px2+y2which lies above thexy-plane inR3. Findthe surface area ofZ.6. The regionDdefined by0.03< x2+y2<1.3isshown at right.Within this region are three curvesA,B, andC. Each curve starts at (0,1) and ends at(0,1). Suppose thatF(x, y) =P(x, y)i+Q(x, y)jis adifferentiable vector field defined onDwith the prop-erties∂P∂y=∂Q∂x,ZAF·dr=1, andZCF·dr= 2.(a) True or false: The regionDis simply connected.(b) True, false, or cannot determine:Fis conservative.(c) FindZBF·dr. Choices:3,2.5,2,1.5,1,0.5, 0, 0.5, 1, 1.5, 2, 2.5, 3Instructor: Matthew C. Russell
7. Show that the line integral2xsiny dx+(x2cosy3y2)dyis independent of the pathC, and evaluateit ifCis any path from (1,0) to (0,2).ZC8. A vector fieldGis plotted at right.(a) Choose the formula forG.

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