Unit 6 of 10

Parametric Equations and Polar Coordinates

  • 17 lessons
  • Parts 145 to 178

Not every curve is the graph of a function y = f(x). Circles, spirals and the paths of moving objects are easier to describe with a parameter such as time, or with a distance and an angle instead of x and y.

This unit covers the conic sections, parametric curves and polar curves, and shows how to find slopes, areas and arc lengths for each of them.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Conics

  1. Definition of a Conic

    What conic sections are, as slices of a double cone and as points whose distances to a focus and a directrix have a fixed ratio, and how eccentricity tells the types apart.

    Theory part 145Practice part 146

  2. Parabolas

    How a parabola is built from a focus and a directrix, how to read its vertex, axis, focus and directrix from its equation, and why it reflects parallel rays into one point.

    Theory part 147Practice part 148

  3. Ellipses

    How an ellipse is defined by a constant sum of distances to two foci, how to find its axes, vertices, foci, eccentricity and directrices from its equation, and why it reflects one focus into the other.

    Theory part 149Practice part 150

  4. Hyperbolas

    How a hyperbola is defined by a constant difference of distances to two foci, how to read its axes, foci, eccentricity, directrices and asymptotes from its equation, and how it reflects light.

    Theory part 151Practice part 152

  5. General Conics

    How to recognize the conic behind any second degree equation in two variables, by completing the square, rotating the axes to remove the cross term, and using the discriminant.

    Theory part 153Practice part 154

Parametric Curves

  1. Parametric Curves

    How a pair of functions of one parameter describes a curve in the plane together with a direction of travel, how to eliminate the parameter, and when a parametrization is smooth.

    Theory part 155Practice part 156

  2. Sketching Parametric Curves

    How to sketch a parametric curve by hand using its parameter interval, endpoints, direction of motion, symmetry and crossing points, and how to eliminate the parameter without adding points that are not on the curve.

    Theory part 157Practice part 158

Calculus of Parametric Curves

  1. The Slope of Parametric Curves

    How to find the slope, tangent and normal lines, horizontal and vertical tangents and concavity of a parametric curve directly from the derivatives of its coordinate functions.

    Theory part 159Practice part 160

  2. Areas Bounded by Parametric Curves

    How the integral of y(t) times x'(t) gives the area under or inside a parametric curve, and how direction, loops and repeated tracing change the sign or the size of the answer.

    Theory part 161Practice part 162

  3. Arc Lengths and Surfaces for Parametric Curves

    How the speed of a moving point gives the length of a parametric curve and the area of the surface it sweeps out when rotated, and when these integrals count a piece twice.

    Theory part 163Practice part 164

Polar Curves

  1. Polar Coordinates

    How to locate a point by its distance from a fixed center and an angle, why the same point has many polar names, and how to switch between polar and rectangular coordinates.

    Theory part 165Practice part 166

  2. Some Polar Curves

    The standard polar curves, circles, cardioids, limaçons, roses and spirals, with how to rotate them, spot their symmetry, find where two of them meet and see how they pass through the pole.

    Theory part 167Practice part 168

  3. Sketching Polar Curves

    A reliable way to sketch a polar curve by hand, using the sign of the radius, symmetry tests, the zeros of the radius and the right interval of angles so that every piece is drawn once.

    Theory part 169Practice part 170

  4. Polar Conics

    How ellipses, parabolas and hyperbolas all come from one focus and directrix rule, and how putting the focus at the pole gives them a single polar equation that shows the eccentricity at a glance.

    Theory part 171Practice part 172

Calculus of Polar Curves

  1. The Slope of Polar Curves

    How to find the slope of a polar curve by treating it as a parametric curve with the angle as parameter, where its tangents are horizontal or vertical, and how to find the tangent lines at the pole.

    Theory part 173Practice part 174

  2. Areas Bounded by Polar Curves

    Why the area swept by a polar curve is one half the integral of r squared, and how to set up areas of loops, petals and regions between two polar curves without counting anything twice.

    Theory part 175Practice part 176

  3. Arc Lengths of Polar Curves

    How the parametric length formula turns into the integral of the square root of r squared plus the square of its derivative, and how to choose the angles so a polar curve is measured exactly once.

    Theory part 177Practice part 178