Calculus Volume 2

# Key Equations

Calculus Volume 2Key Equations
• Area between two curves, integrating on the x-axis
$A=∫ab[f(x)−g(x)]dxA=∫ab[f(x)−g(x)]dx$
• Area between two curves, integrating on the y-axis
$A=∫cd[u(y)−v(y)]dyA=∫cd[u(y)−v(y)]dy$
• Disk Method along the x-axis
$V=∫abπ[f(x)]2dxV=∫abπ[f(x)]2dx$
• Disk Method along the y-axis
$V=∫cdπ[g(y)]2dyV=∫cdπ[g(y)]2dy$
• Washer Method
$V=∫abπ[(f(x))2−(g(x))2]dxV=∫abπ[(f(x))2−(g(x))2]dx$
• Method of Cylindrical Shells
$V=∫ab(2πxf(x))dxV=∫ab(2πxf(x))dx$
• Arc Length of a Function of x
$Arc Length=∫ab1+[f′(x)]2dxArc Length=∫ab1+[f′(x)]2dx$
• Arc Length of a Function of y
$Arc Length=∫cd1+[g′(y)]2dyArc Length=∫cd1+[g′(y)]2dy$
• Surface Area of a Function of x
$Surface Area=∫ab(2πf(x)1+(f′(x))2)dxSurface Area=∫ab(2πf(x)1+(f′(x))2)dx$
• Mass of a one-dimensional object
$m=∫abρ(x)dxm=∫abρ(x)dx$
• Mass of a circular object
$m=∫0r2πxρ(x)dxm=∫0r2πxρ(x)dx$
• Work done on an object
$W=∫abF(x)dxW=∫abF(x)dx$
• Hydrostatic force on a plate
$F=∫abρw(x)s(x)dxF=∫abρw(x)s(x)dx$
• Mass of a lamina
$m=ρ∫abf(x)dxm=ρ∫abf(x)dx$
• Moments of a lamina
$Mx=ρ∫ab[f(x)]22dxandMy=ρ∫abxf(x)dxMx=ρ∫ab[f(x)]22dxandMy=ρ∫abxf(x)dx$
• Center of mass of a lamina
$x–=Mymandy–=Mxmx–=Mymandy–=Mxm$
• Natural logarithm function
• $lnx=∫1x1tdtlnx=∫1x1tdt$ Z
• Exponential function $y=exy=ex$
• $lny=ln(ex)=xlny=ln(ex)=x$ Z