Calculus Volume 3

Key Equations

Calculus Volume 3Key Equations

Key Equations

 Double integral $∬Rf(x,y)dA=limm,n→∞∑i=1m∑j=1nf(xij*,yij*)ΔA∬Rf(x,y)dA=limm,n→∞∑i=1m∑j=1nf(xij*,yij*)ΔA$ Iterated integral $∫ab∫cdf(x,y)dxdy=∫ab[∫cdf(x,y)dy]dx∫ab∫cdf(x,y)dxdy=∫ab[∫cdf(x,y)dy]dx$ or $∫cd∫baf(x,y)dxdy=∫cd[∫abf(x,y)dx]dy∫cd∫baf(x,y)dxdy=∫cd[∫abf(x,y)dx]dy$ Average value of a function of two variables $fave=1AreaR∬Rf(x,y)dxdyfave=1AreaR∬Rf(x,y)dxdy$
 Iterated integral over a Type I region $∬Df(x,y)dA=∬Df(x,y)dydx=∫ab[∫g1(x)g2(x)f(x,y)dy]dx∬Df(x,y)dA=∬Df(x,y)dydx=∫ab[∫g1(x)g2(x)f(x,y)dy]dx$ Iterated integral over a Type II region $∬Df(x,y)dA=∬Df(x,y)dxdy=∫cd[∫h1(y)h2(y)f(x,y)dx]dy∬Df(x,y)dA=∬Df(x,y)dxdy=∫cd[∫h1(y)h2(y)f(x,y)dx]dy$
 Double integral over a polar rectangular region $RR$ $∬Rf(r,θ)dA=limm,n→∞∑i=1m∑j=1nf(rij*,θij*)ΔA=limm,n→∞∑i=1m∑j=1nf(rij*,θij*)rij*ΔrΔθ∬Rf(r,θ)dA=limm,n→∞∑i=1m∑j=1nf(rij*,θij*)ΔA=limm,n→∞∑i=1m∑j=1nf(rij*,θij*)rij*ΔrΔθ$ Double integral over a general polar region $∬Df(r,θ)rdrdθ=∫θ=αθ=β∫r=h1(θ)r=h2(θ)f(r,θ)rdrdθ∬Df(r,θ)rdrdθ=∫θ=αθ=β∫r=h1(θ)r=h2(θ)f(r,θ)rdrdθ$
 Triple integral $liml,m,n→∞∑i=1l∑j=1m∑k=1nf(xijk*,yijk*,zijk*)ΔxΔyΔz=∭Bf(x,y,z)dVliml,m,n→∞∑i=1l∑j=1m∑k=1nf(xijk*,yijk*,zijk*)ΔxΔyΔz=∭Bf(x,y,z)dV$
 Triple integral in cylindrical coordinates $∭Bg(x,y,z)dV=∭Bg(rcosθ,rsinθ,z)rdrdθdz=∭Bf(r,θ,z)rdrdθdz∭Bg(x,y,z)dV=∭Bg(rcosθ,rsinθ,z)rdrdθdz=∭Bf(r,θ,z)rdrdθdz$ Triple integral in spherical coordinates $∭Bf(ρ,θ,φ)ρ2sinφdρdφdθ=∫φ=γφ=ψ∫θ=αθ=β∫ρ=aρ=bf(ρ,θ,φ)ρ2sinφdρdφdθ∭Bf(ρ,θ,φ)ρ2sinφdρdφdθ=∫φ=γφ=ψ∫θ=αθ=β∫ρ=aρ=bf(ρ,θ,φ)ρ2sinφdρdφdθ$
 Mass of a lamina $m=limk,l→∞∑i=1k∑j=1lmij=limk,l→∞∑i=1k∑j=1lρ(xij*,yij*)ΔA=∬Rρ(x,y)dAm=limk,l→∞∑i=1k∑j=1lmij=limk,l→∞∑i=1k∑j=1lρ(xij*,yij*)ΔA=∬Rρ(x,y)dA$ Moment about the x-axis $Mx=limk,l→∞∑i=1k∑j=1l(yij*)mij=limk,l→∞∑i=1k∑j=1l(yij*)ρ(xij*,yij*)ΔA=∬Ryρ(x,y)dAMx=limk,l→∞∑i=1k∑j=1l(yij*)mij=limk,l→∞∑i=1k∑j=1l(yij*)ρ(xij*,yij*)ΔA=∬Ryρ(x,y)dA$ Moment about the y-axis $My=limk,l→∞∑i=1k∑j=1l(xij*)mij=limk,l→∞∑i=1k∑j=1l(xij*)ρ(xij*,yij*)ΔA=∬Rxρ(x,y)dAMy=limk,l→∞∑i=1k∑j=1l(xij*)mij=limk,l→∞∑i=1k∑j=1l(xij*)ρ(xij*,yij*)ΔA=∬Rxρ(x,y)dA$ Center of mass of a lamina $x−=Mym=∬Rxρ(x,y)dA∬Rρ(x,y)dAx−=Mym=∬Rxρ(x,y)dA∬Rρ(x,y)dA$ and $y−=Mxm=∬Ryρ(x,y)dA∬Rρ(x,y)dAy−=Mxm=∬Ryρ(x,y)dA∬Rρ(x,y)dA$
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