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Calculus Volume 1

Key Equations

Calculus Volume 1Key Equations

Key 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=ρabf(x)dxm=ρabf(x)dx
Moments of a lamina Mx=ρab[f(x)]22dxandMy=ρabxf(x)dxMx=ρab[f(x)]22dxandMy=ρabxf(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
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