Lecture Note: §11 Parametric Equations and Polar Coordinates

Last Update: 2026-03-22

§11.1 Parametrizations of Plane Curves

• Parametric Curve (Equations):
  • $x = f(t), \ y = g(t)$

• Cycloids example

• Graphing parametric curves

§11.2 Calculus with Parametric Curves

• Tangents:
  • $\displaystyle \frac{d y}{d x}=\frac{d y / d t}{d x / d t}$
  • $\displaystyle \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(y^{\prime}\right)=\frac{d y^{\prime} / d t}{d x / d t}$
• Length of a Parametrically Defined Curve
  • $\displaystyle L=\int_a^{b} \sqrt{\left[f^{\prime}(t)\right]^{2}+\left[g^{\prime}(t)\right]^{2}}\, dt.$
  • The Arc Length Differential $\displaystyle ds=\sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}}\, dt$

• Area Under the Curve

• Area of Surfaces of Revolution
  • Revolution about x-axis: $\displaystyle S=\int_a^{b} 2 \pi y \sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2} d t}$
  • Revolution about y-axis: $\displaystyle S=\int_a^{b} 2 \pi x \sqrt{\left(\frac{d x}{d t}\right)^2+\left(\frac{d y}{d t}\right)^{2} d t}$

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§11.3 Polar Coordinates

• Definition of Polar Coordinates $P(r, \theta)$
  • $r$: Directed distance from $O$ to $P$
  • $\theta$: Directed angle from the initial ray to $OP$

• Equations Relating Polar and Cartesian Coordinates

  • Polar to Cartesian: $x=r \cos \theta, \quad y=r \sin \theta$
  • Cartesian to Polar: $r^{2}=x^{2}+y^{2}, \quad \tan \theta=\dfrac{y}{x}$
  • Some curves are more simply expressed with polar coordinates, such as cardioid, etc.

• Slope of the Curve $r = f(\theta)$ in the Cartesian $xy$-Plane

  • $x=r \cos \theta=f(\theta) \cos \theta, \, y=r \sin \theta=f(\theta) \sin \theta .$

  • $\displaystyle \left.\frac{d y}{d x}\right|_{(r, \theta)}=\frac{f^{\prime}(\theta) \sin \theta+f(\theta) \cos \theta}{f^{\prime}(\theta) \cos \theta-f(\theta) \sin \theta}$

§11.4 Graphing Polar Coordinate Equations

§11.5 Areas and Lengths in Polar Coordinates

• Area of the Fan-Shaped Region: $\displaystyle A=\int_\alpha^{\beta} \frac{1}{2} r^{2}\, d\theta$.

• Length of a Polar Curve $\displaystyle L=\int_\alpha^{\beta} \sqrt{r^{2}+\left(\frac{d r}{d \theta}\right)^2}\, d\theta$

§11.6 Conic Sections

• Parabolas

• Ellipses

• Hyperbolas

§11.7 Conics in Polar Coordinates

• A Unified Description of Conics
  • $\displaystyle \text{Eccentricity} = \frac{\text{distance between foci}}{\text{distance between vertices}}$
  • Suppose that the distance $PF$ of a point $P$ from a fixed point $F$ (the focus) is a constant multiple of its distance from a fixed line (the directrix). That is, suppose \(PF = e\cdot PD\) where is the constant of proportionality. Then the path traced by $P$is
    1. a parabola if $e = 1$,
    2. an ellipse of eccentricity $e$ if $e < 1$, and
    3. a hyperbola of eccentricity $e$ if $e > 1$.
• Polar Equation for a Conic with Eccentricity $e$
  1. $\displaystyle r=\frac{k e}{1 + e \cos \theta}\quad\text{or}\quad $ where $x=k>0$ is the vertical directrix.
  2. $\displaystyle r=\frac{k e}{1 - e \cos \theta}\quad\text{or}\quad $ where $x=-k<0$ is the vertical directrix.
  3. $\displaystyle r=\frac{k e}{1 + e \sin \theta}\quad\text{or}\quad $ where $y=k>0$ is the horizontal directrix.
  4. $\displaystyle r=\frac{k e}{1 - e \sin \theta}\quad\text{or}\quad $ where $y=-k<0$ is the horizontal directrix.
• The Standard Polar Equation for Lines. If the point $P_0(r_0,\theta_0)$ is the foot of the perpendicular from the origin to the line $L$, and $r_0\geq 0$, then an equation for $L$ is $r \cos(\theta - \theta_0) = r_0$.