Calculus Volume 2

# Key Equations

Calculus Volume 2Key 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
$L=∫αβ[f(θ)]2+[f′(θ)]2dθ=∫αβr2+(drdθ)2dθL=∫αβ[f(θ)]2+[f′(θ)]2dθ=∫αβr2+(drdθ)2dθ$