HW13-solutions - jiang(xj842 HW13 allen(54060 This...

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jiang (xj842) – HW13 – allen – (54060)1Thisprint-outshouldhave16questions.Multiple-choice questions may continue onthe next column or page – find all choicesbefore answering.00110.0pointsUse the transformation (x, y)(u, v) withx= 4u ,y= 3vto evaluate the integralI=integraldisplay integraldisplayRx2dxdywhenRis the region bounded by the ellipsex216+y29= 1.1.I= 12π2.I= 33.I= 484.I= 48πcorrect5.I= 3π6.I= 12Explanation:The transformation (x, y)(u, v) withx= 4u ,y= 3vmaps the elliptical diskRxy=braceleftBig(x, y) :x216+y291bracerightBigin thexy-plane into the circular diskRuv=braceleftBig(u, v) :u2+v21bracerightBigof radius 1 in theuv-plane. Since the Jacobianof this transformation is given by(x, y)(u, v)= 12,the integralI=integraldisplay integraldisplayRxyx2dxdythus becomesI= 12integraldisplay integraldisplayRuv(4u)2dudvafter the change of variable fromx, ytou, v.But this last integral is of polar type, so toevaluate we next change to polar coordinates:I= 192integraldisplay10integraldisplay2π0r2cos2θ(rdθdr)= 192integraldisplay10integraldisplay2π0r3cos2θ dθdr .Butintegraldisplay2π0cos2θ dθ=12integraldisplay2π0(1 + cos 2θ)=12bracketleftBigθ+12sin 2θbracketrightBig2π0=π .Consequently,I= 192πintegraldisplay10r3dr= 48π.00210.0pointsBy using an appropriate transformation,evaluate the integralI=integraldisplay integraldisplayRparenleftBigx236+y216parenrightBig1/2dxdywhenRis the region enclosed by the graph ofx236+y216= 1.1.I= 12π2.I= 24
jiang (xj842) – HW13 – allen – (54060)23.I= 24π4.I= 16πcorrect5.I= 126.I= 16Explanation:The transformation (x, y)(u, v) withx= 6u ,y= 4vmaps the elliptical diskRxy=braceleftBig(x, y) :x236+y2161bracerightBigin thexy-plane into the circular diskRuv=braceleftBig(u, v) :u2+v21bracerightBigof radius 1 in theuv-plane. Since the Jacobianof this transformation is given by(x, y)(u, v)= 24,the integral thus becomesI= 24integraldisplay integraldisplayRuv(u2+v2)1/2dudvafter the change of variable fromx, ytou, v.But this last integral is of polar type, so toevaluate we next change to polar coordinates:I= 24integraldisplay10integraldisplay2π0r2dθdr .Consequently,I= 16π.00310.0pointsEvaluate the integralI=integraldisplay integraldisplayDx-3y2x-ydAwhenDis the parallelogram bounded byx-3y= 0,x-3y= 2,and2x-y= 1,2x-y= 3,by making an appropriate change of variables.1.I=252.I=45ln 33.I= 04.I=25ln 3correct5.I=45Explanation:Settingu=x-3y ,v= 2x-ysimplifies both the integrand and the regionof integrationD. To determine the change ofvariableT: (u, v)(x, y)we first need to solve forx, yin terms ofu, v:x=15(3v-u),y=15(v-2v).

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