Lecture 9 Notes: Arc Length and Polar Coordinates

Context and Course Logistics

• Announcement: Week 6 has no workshops; mid-semester exam on Monday; quiz extended to Thursday due to the Monday exam.
• Topic focus: Arc length and polar coordinates; polar curves are closely related to parametric curves and can be viewed as a special type of parametric curve.

Parametric Curves: Key Concepts Recap

• How to plot parametric curves: identify tangents, horizontal/vertical tangents, multiple tangents, and test for concavity using the second-derivative test.
• Arc length for parametric curves: the arc length is the integral of speed along the curve.
• Speed interpretation: treat the parametric path as the trajectory of a particle; at time t the velocity vector is
  \mathbf{v}(t) = (x'(t), y'(t)) and its speed is the magnitude |\mathbf{v}(t)| = \sqrt{(x'(t))^2 + (y'(t))^2}.
• Arc length formula: for a parameter t in ([\alpha, \beta]),
  L = \int_{\alpha}^{\beta} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt.
• Example approach: compute x'(t) and y'(t), set the limits (\alpha, \beta) (e.g., 0 to 2), and form the integral; some integrals are not elementary and may require numerical methods.
• Numerical methods: Riemann sums, left/right endpoints, trapezoid rule; software (e.g., Wolfram Alpha) can provide numerical approximations (e.g., decimal results).

Example: Arc Length for a Parametric Curve

• Given a parametric curve with
  x'(t) = 3t^2 - 3,
  \quad y'(t) = 2t,\qquad t \in [0,2],
  the arc length is
  L = \int{0}^{2} \sqrt{(3t^2-3)^2 + (2t)^2} \, dt = \int{0}^{2} \sqrt{9t^4 - 14t^2 + 9} \, dt.
• This integral is typically hard to evaluate in elementary closed form; numerical evaluation yields approximately
  L \approx 7.0461.
• Conceptual takeaway: arc length is the accumulated speed along the path; if the integral is difficult, numerical methods are standard in practice.

The Cycloid: Parametrization, Arc Length, and Area

• Physical picture: A wheel of radius r rolls without slipping; a fixed point on the rim traces a cycloid.
• Standard parametric equations for one arch (θ from 0 to 2π):
  x(\theta) = r(\theta - \sin\theta),\qquad y(\theta) = r(1 - \cos\theta).
• Derivation idea: the horizontal translation equals the arc length rolled (proportional to θ), while the vertical motion comes from the rotation of the wheel.
• One arch corresponds to θ ∈ [0, 2π].
• Area under one arch (parametric area): for a region under a parametric curve, the area can be computed via
  A = \int y \, dx = \int y(t)\, x'(t) \, dt,
  where x and y are given parametrically by x = f(t), y = g(t).
• For the cycloid, with the above x and y, we have
  x'(\theta) = r(1 - \cos\theta),\quad y(\theta) = r(1 - \cos\theta).
• Therefore, the area under one arch is
  A = \int{0}^{2\pi} y(\theta) x'(\theta) \, d\theta = \int{0}^{2\pi} r^2(1 - \cos\theta)^2 \, d\theta.
• Expanding and integrating:
  A = r^2 \int{0}^{2\pi} \left(1 - 2\cos\theta + \cos^2\theta\right) d\theta = r^2 \left[ \int{0}^{2\pi} 1 \, d\theta - 2\int{0}^{2\pi} \cos\theta \, d\theta + \int{0}^{2\pi} \cos^2\theta \, d\theta \right].
  Using (\cos^2\theta = \frac{1 + \cos 2\theta}{2}) and the fact that the integrals of (\cos\theta) and (\cos 2\theta) over one full period vanish, we get
  A = r^2 \left( 2\pi - 0 + \pi \right) = 3\pi r^2.
• Key takeaway: area under one arch of a cycloid is (A = 3\pi r^2).

Parametric Curves in Higher Dimensions: Helix and Coordinate Curves

• Helix example: a circle in the xy-plane with a linear rise in z.
  • Parametrization: x(t) = \cos t,\quad y(t) = \sin t,\quad z(t) = t, with t typically real.
  • For three full turns, t goes from 0 to (6\pi).
  • Velocity: \mathbf{v}(t) = (-\sin t, \cos t, 1), so the speed is
    |\mathbf{v}(t)| = \sqrt{(-\sin t)^2 + (\cos t)^2 + 1^2} = \sqrt{2}.
  • Arc length for three turns:
    L = \int_{0}^{6\pi} \sqrt{2} \, dt = 6\pi\sqrt{2}.
• Coordinate curves on a paraboloid (a 3D surface): consider a paraboloid with equation e.g. z = x^2 + y^2.
  • A coordinate curve can be generated by fixing a direction in the xy-plane, described by a unit vector (\mathbf{u} \in \mathbb{R}^2) and letting t run along that direction while z follows the surface.
  • If (\mathbf{u} = (ux, uy)) with (ux^2 + uy^2 = 1), a coordinate curve is parameterized by
    x(t) = t ux,\quad y(t) = t uy,\quad z(t) = x(t)^2 + y(t)^2 = t^2.
  • Example: take (\mathbf{u} = \left( \tfrac{1}{\sqrt{3}}, \tfrac{\sqrt{2}}{3} \right)) (which is unit-length). Then
    x(t) = \frac{t}{\sqrt{3}},\quad y(t) = t\cdot \frac{\sqrt{2}}{3},\quad z(t) = t^2,
    and t runs over the real numbers.
• Practical note: these coordinate curves help build geometric intuition in higher dimensions; partial derivatives along these curves give insight into surface behavior.

Polar Coordinates: Basics and Why They Help

• Polar coordinates describe a point in the plane using radius r and angle θ: the same point satisfies
  • Cartesian relations: x = r\cos\theta,\quad y = r\sin\theta,\quad r^2 = x^2 + y^2.
  • Inverse relations: \theta = \operatorname{arctan}\left(\frac{y}{x}\right) (with quadrant awareness).
• Relationship between the two coordinate systems: rotating by π changes the sign of r, i.e., (−r, θ) is the same as (r, θ+π).
• Conversion back and forth relies on basic trigonometry and the identity \sin^2\theta + \cos^2\theta = 1.
• Why polar can be simpler: radial or symmetric shapes are often easier to describe in polar form; many curves are straightforward in polar form but more involved in Cartesian form, and vice versa for some lines and circles.

Examples in Polar Coordinates: From Polar to Cartesian and Back

• Circle via r = 2: this is a circle of radius 2 centered at the origin; in Cartesian form it’s simply x^2 + y^2 = 4.
• Line through the origin with a fixed angle: θ = φ describes a straight line through the origin making angle φ with the positive x-axis.
• Cardioid: r = 1 + \sin\theta
  • Plot strategy: first plot r as a function of θ (a 1D plot), then interpret the polar points ((r, \theta)) as Cartesian points via x = r cos θ, y = r sin θ.
  • Key points: at θ = 0, r = 1 ⇒ (x,y) = (1,0); at θ = π/2, r = 2 ⇒ (x,y) = (0,2); at θ = π, r = 1 ⇒ (x,y) = (−1,0); at θ = 3π/2, r = 0 ⇒ (0,0) (the origin).
  • The plotted path forms a heart-shaped cardioid as θ goes from 0 to 2π.
• Rose curves: r = 2 cos(2θ) yields a four-leaved rose.
  • Explanation: replacing θ by 2θ increases the angular frequency, producing more oscillations (petals) within 0 to 2π; the amplitude is doubled, so petals reach out to r = 2 and cross the origin multiple times.
  • Plotting approach mirrors the cardioid example, but with more frequent crossings of the origin due to the factor 2 in the cosine argument.

Plotting Polar Curves: Practical Tips

• Plot the function r = f(θ) first to understand how far out the curve extends for given angles.
• Convert to Cartesian when helpful to recognize the shape (e.g., circles, lines, ellipses) by using x = r cos θ and y = r sin θ together with r^2 = x^2 + y^2.
• Be mindful of the origin crossings: r = 0 at certain θ values means the curve passes through the origin; count how often this occurs to understand the shape (e.g., in roses).
• Polar plots can reveal symmetry and radial structure that are not immediately obvious from Cartesian plots.

Connections and Implications

• Relationship to calculus in higher dimensions: polar coordinates provide a bridge to analyzing curves and surfaces with symmetry; they complement the parametric perspective by offering alternative descriptions that can simplify differentiation, integration, and geometric interpretation.
• Real-world relevance: arc length calculations model actual distances traveled; area under curves in parametric form extends to surfaces and higher-dimensional analogues via appropriate coordinate systems.
• Conceptual takeaways: drawing and geometric intuition are emphasized for higher-dimensional thinking; parameterizations expose how motion and geometry intertwine (velocity, speed, and area under parametric graphs).

Quick Reference Formulas (LaTeX)

• Arc length for a parametric curve:
  L = \int_{\alpha}^{\beta} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt.
• Velocity vector and speed:
  \mathbf{v}(t) = \bigl(x'(t), y'(t)\bigr),\quad |\mathbf{v}(t)| = \sqrt{(x'(t))^2 + (y'(t))^2}.
• Cycloid parametrics (one arch):
  x(\theta) = r(\theta - \sin\theta),\quad y(\theta) = r(1 - \cos\theta),\qquad \theta \in [0, 2\pi].
• Area under one arch of cycloid:
  A = \int{0}^{2\pi} y(\theta) x'(\theta) \, d\theta = \int{0}^{2\pi} r^2(1 - \cos\theta)^2 \, d\theta = 3\pi r^2.
• Helix parametrization and arc length for three turns:
  x(t) = \cos t,\quad y(t) = \sin t,\quad z(t) = t,
  |\mathbf{v}(t)| = \sqrt{(-\sin t)^2 + (\cos t)^2 + 1^2} = \sqrt{2},
  \quad L = \int_{0}^{6\pi} \sqrt{2} \, dt = 6\pi\sqrt{2}.$$