Lec4_New.pdf - Calculus II Lecture 4 Outline 1 A brief...

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Calculus IILecture 4
Outline1A brief recall of Lecture 32Double integral over general domain3Integrals in Polar Coordinates4Application of double integralDensity and MassProbability5Surface area6Triple Integrals (15.7)
Double integralLetfbe a function of two variables defined onR= [a,b]×[c,d] ={(x,y) :x[a,b],y[c,d]}.LetSbe the solid aboveRand under the graphf:S:={(x,y,z) :0zf(x,y),(x,y)R}.Find the volume ofS.
IdeaSubrectanglesΔA= ΔxΔy
IdeaV=mi=1nj=1f(x*ij,y*ijA.RRRf(x,y)dA=limm,n→∞mi=1nj=1f(x*ij,y*ijA.
Interated integralsLetfbe a function defined on[a,b]×[c,d].LetA(x) :=Rdcf(x,y)dyThe integralZbaA(x)dx=ZbaZdcf(x,y)dydxis called aniterated integral.Fubini’s TheoremZZRf(x,y)dA=Zba(Zdcf(x,y)dy)dx=Zdc(Zbaf(x,y)dx)dy.In particular,ZZRg(x)h(y)dA=Zbag(x)dxZdch(y)dy.

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