CALC 2: PARAMETRIC & POLAR

PARAMETRIC CURVES
PARAMETRIC CALCULUS
POLAR COORDINATES
POLAR AREAS & LENGTHS
PARAMETRIC EQUATIONS
x = f(t)
y = g(t)

where t is the parameter
CONVERSION FORMULAS
Parametric → Cartesian:
Eliminate t from both equations

Example: x = t², y = 2t
t = y/2, so x = (y/2)² = y²/4
t = 6.28
Speed: 1.0x
CURRENT CURVE: Circle
EQUATIONS: x = cos(t), y = sin(t)
TANGENT LINE
dy/dx = (dy/dt)/(dx/dt)

Tangent line at t = t₀:
y - y₀ = (dy/dx)|ₜ₌ₜ₀ · (x - x₀)
ARC LENGTH
L = ∫ᵃᵇ √[(dx/dt)² + (dy/dt)²] dt

where a ≤ t ≤ b
t₀ = 1.57
TANGENT SLOPE: 0
ARC LENGTH (0 to t₀): 0
POLAR COORDINATES
r = distance from origin
θ = angle from positive x-axis

Point: (r, θ)
CONVERSION FORMULAS
Polar → Cartesian:
x = r cos(θ)
y = r sin(θ)

Cartesian → Polar:
r = √(x² + y²)
θ = arctan(y/x)
θ = 6.28
CURRENT CURVE: Circle
EQUATION: r = 2
AREA IN POLAR
A = (1/2) ∫ᵃᵇ r² dθ

where α ≤ θ ≤ β
ARC LENGTH IN POLAR
L = ∫ᵃᵇ √[r² + (dr/dθ)²] dθ

where α ≤ θ ≤ β
α = 0
β = 3.14
AREA: 0
ARC LENGTH: 0