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    • ▾Calculus 2
      • ▸Integration techniques
        • •Intro to calc 2
        • ▸Basic indefinite integrals
          • •Reverse power rule for indefinite integrals
          • •Sum, difference, and reverse power rules for indefinite integrals
        • •Integration by parts
        • ▸Integrals involving trig functions
          • •Lesson
          • •Practice
        • •Trig substitutions
        • •Integration by partial fractions
        • •Integral recurrence relations
        • •Integrating quadratics by completing the square
        • •Integration by long division
        • •Improper integrals
        • •Comparison tests for convergence
        • ▸Approximating definite integrals
          • •Riemann sums
          • ▸Simpson's rule
            • •Practice: Simpson's rule
          • ▸Trapezoidal rule
            • •Lesson
            • •Practice
          • •Midpoint rule
      • ▸Applications of integrals
        • ▸Arc length (calculus 2)
          • •Using the formula
        • ▸Center of mass (calculus 2)
          • •Center of mass for one and two-dimensional systems
          • •Center of mass for planar lamina
          • •Pappus's theorem
        • •Hydrostatic force
        • •Mode of a continuous random variable from a probability density function
        • •Surface area of solids of revolution
        • •Gabriel's Horn
        • •Applications of trigonometric integrals
        • •Integrating exponential functions
        • •Logarithmic functions
      • ▾Parametric equations and polar coordinates
        • •Cycloid area and length
        • •Eliminating the parameter
        • •Testing polar equations for symmetry
        • •Deriving the equations of a cycloid, epicycloid, and hypocycloid
      • ▸Series and sequences
        • •Maclaurin series
        • •Finding the sum of a telescoping series
        • ▸Convergent and divergent geometric series
          • •Lesson
          • •Practice
        • •Alternating harmonic series
        • •Alternating series test
        • •Partial sums
        • •Ratio test
        • ▸Root test
          • •Lesson
          • •Practice
        • •Absolute convergence, conditional convergence, and divergence
        • •Riemann's rearrangement theorem
        • •Taylor series
      • ▸Vectors
        • •Intro to vectors
        • •Length of a vector
        • •Cross product
        • •Intro to the dot product
      • •Competition problems (calculus 2)
     › Calculus 2 › Parametric equations and polar coordinates

    Deriving the equations of a cycloid, epicycloid, and hypocycloid

    First, students will learn two ways of deriving the equations of a cycloid.

    Watch these videos:

    • Tim Hodges

      • Deriving the Equations of a Cycloid by Xander Gouws

    Next, students will learn how to derive the equations of an epicycloid.

    Deriving the Equations of an Epicycloid by Xander Gouws

    After that, students will learn how to derive the equations of a hypocycloid.

    Hypocycloid derivation of parametric equations and examples of hypocycloid animations. by Zak's Lab

    Conclude by giving your students these challenges:

    • Spot the Card by NRICH
    • Factorising with Multilink by NRICH
    • Concrete Calculation by NRICH
    • 2005 AMC 8, Problem 12
    • Chords in a Circle

    Draw four circles, such that each circle contains an odd number of roses, each circle contains a different number of roses, and no two circles touch or intersect:

    Here's the solution:

    Source: Test Your Math IQ by Steve Ryan