Calculus Applied to Parametrically-Defined Curves

  1. Consider the closed bowtie-shaped curve defined parametrically for \(t\) between \(0\) and \(2\pi\) by the coordinates \[x(t) = \sin(t) \quad\text{and}\quad y(t) = \sin(t)\cos(t)\,.\]
    1. Confirm that the initial point and terminal point of this curve segment are the same point, making it a closed curve.
    2. What is an equation for the line tangent to the curve at the point where \(t = \pi/3\)?
    3. What is the area of the bowtie-shaped region enclosed by the curve?
    4. Write down an integral that computes the length of this curve.
    5. Write down an integral that computes the surface area of the solid generated by revolving the region enclosed by the curve about the \(x\)-axis?
  2. Consider the cycloid curve defined parametrically by \[x(t) = 3\big(t-\sin(t)\big) \quad\text{and}\quad y(t) = 3\big(1-\cos(t)\big)\,.\]
    1. Calculate the slope of the line tangent to the cycloid at the point where \(t = \pi/3\).
    2. At what points on the cycloid is the tangent line perfectly horizontal? At what points on the cycloid is the tangent line perfectly vertical?
    3. Calculate the area of the region under one “arch” of the cycloid
    4. Calculate the length of a single “arch” of the cycloid.