| |

1: Parametric Equations / Polar Coordinates

Alternative coordinate systems for describing curves and regions.

Parametric Equations

Instead of y=f(x)y = f(x), we describe a curve using a parameter tt:

x=f(t),y=g(t),a≤t≤bx = f(t), \quad y = g(t), \quad a \leq t \leq b

Why Parametric?

  • Can describe curves that aren’t functions (e.g., circles)
  • Natural for motion: tt represents time
  • Can trace the same curve in different directions/speeds

Examples

Circle of radius rr:

x=rcos⁡t,y=rsin⁡t,0≤t≤2πx = r\cos t, \quad y = r\sin t, \quad 0 \leq t \leq 2\pi

Ellipse with semi-axes aa and bb:

x=acos⁡t,y=bsin⁡t,0≤t≤2πx = a\cos t, \quad y = b\sin t, \quad 0 \leq t \leq 2\pi

Cycloid (point on a rolling wheel):

x=r(t−sin⁡t),y=r(1−cos⁡t)x = r(t - \sin t), \quad y = r(1 - \cos t)

Calculus with Parametric Curves

Derivatives

If x=f(t)x = f(t) and y=g(t)y = g(t), the slope is:

dydx=dy/dtdx/dt=g′(t)f′(t)\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}

Second derivative:

d2ydx2=ddx(dydx)=ddt(dydx)dx/dt\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{dx/dt}

Arc Length

The length of a parametric curve from t=at = a to t=bt = b:

L=∫ab(dxdt)2+(dydt)2 dtL = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt

Area Under a Parametric Curve

A=∫aby dx=∫abg(t)f′(t) dtA = \int_a^b y \, dx = \int_a^b g(t) f'(t) \, dt

Polar Coordinates

A point is described by distance from origin and angle:

(r,θ)where r≥0,0≤θ<2π(r, \theta) \quad \text{where } r \geq 0, \quad 0 \leq \theta < 2\pi

Conversion Formulas

Polar to Cartesian:

x=rcos⁡θ,y=rsin⁡θx = r\cos\theta, \quad y = r\sin\theta

Cartesian to Polar:

r=x2+y2,tan⁡θ=yxr = \sqrt{x^2 + y^2}, \quad \tan\theta = \frac{y}{x}

Common Polar Curves

CurveEquation
Circle (radius aa)r=ar = a
Circle through originr=acos⁡θr = a\cos\theta or r=asin⁡θr = a\sin\theta
Line through originθ=c\theta = c
Cardioidr=a(1+cos⁡θ)r = a(1 + \cos\theta)
Rose (nn petals)r=acos⁡(nθ)r = a\cos(n\theta)
Lemniscater2=a2cos⁡(2θ)r^2 = a^2\cos(2\theta)
Spiralr=aθr = a\theta

Calculus in Polar Coordinates

Slope of a Polar Curve

Since x=rcos⁡θx = r\cos\theta and y=rsin⁡θy = r\sin\theta:

dydx=dy/dθdx/dθ=drdθsin⁡θ+rcos⁡θdrdθcos⁡θ−rsin⁡θ\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = \frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta}

Area in Polar Coordinates

Area enclosed by r=f(θ)r = f(\theta) from θ=α\theta = \alpha to θ=β\theta = \beta:

A=12∫αβr2 dθ=12∫αβ[f(θ)]2 dθA = \frac{1}{2}\int_\alpha^\beta r^2 \, d\theta = \frac{1}{2}\int_\alpha^\beta [f(\theta)]^2 \, d\theta

Intuition: A thin wedge has area 12r2 dθ\frac{1}{2}r^2 \, d\theta (like a pizza slice).

Arc Length in Polar Coordinates

L=∫αβr2+(drdθ)2 dθL = \int_\alpha^\beta \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta

Summary

ConceptCartesianParametricPolar
Curvey=f(x)y = f(x)x=f(t),y=g(t)x = f(t), y = g(t)r=f(θ)r = f(\theta)
Slopedydx\frac{dy}{dx}dy/dtdx/dt\frac{dy/dt}{dx/dt}r′sin⁡θ+rcos⁡θr′cos⁡θ−rsin⁡θ\frac{r'\sin\theta + r\cos\theta}{r'\cos\theta - r\sin\theta}
Arc length∫1+(dy/dx)2 dx\int\sqrt{1 + (dy/dx)^2}\,dx∫(dx/dt)2+(dy/dt)2 dt\int\sqrt{(dx/dt)^2 + (dy/dt)^2}\,dt∫r2+(dr/dθ)2 dθ\int\sqrt{r^2 + (dr/d\theta)^2}\,d\theta
Area∫y dx\int y\,dx∫g(t)f′(t) dt\int g(t)f'(t)\,dt12∫r2 dθ\frac{1}{2}\int r^2\,d\theta