Calculus Volume 2

# Key Equations

Calculus Volume 2Key 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 $m=ρ∫abf(x)dxm=ρ∫abf(x)dx$ 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