Calculus Volume 3

# Key Equations

Calculus Volume 3Key Equations

### Key Equations

 Derivative of parametric equations $dydx=dy/dtdx/dt=y′(t)x′(t)dydx=dy/dtdx/dt=y′(t)x′(t)$ Second-order derivative of parametric equations $d2ydx2=ddx(dydx)=(d/dt)(dy/dx)dx/dtd2ydx2=ddx(dydx)=(d/dt)(dy/dx)dx/dt$ Area under a parametric curve $A=∫aby(t)x′(t)dtA=∫aby(t)x′(t)dt$ Arc length of a parametric curve $s=∫t1t2(dxdt)2+(dydt)2dts=∫t1t2(dxdt)2+(dydt)2dt$ Surface area generated by a parametric curve $S=2π∫aby(t)(x′(t))2+(y′(t))2dtS=2π∫aby(t)(x′(t))2+(y′(t))2dt$
 Area of a region bounded by a polar curve $A=12∫αβ[f(θ)]2dθ=12∫αβr2dθA=12∫αβ[f(θ)]2dθ=12∫αβr2dθ$ Arc length of a polar curve $L=∫αβ[f(θ)]2+[f′(θ)]2dθ=∫αβr2+(drdθ)2dθL=∫αβ[f(θ)]2+[f′(θ)]2dθ=∫αβr2+(drdθ)2dθ$
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