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Calculus Volume 3

# Key Equations

Calculus Volume 3Key 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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