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Step 16: Calculus 2

Parametric and polar calculus

Step 16 · Calculus 2 topics

Find slopes and lengths of curves given by x(t) and y(t), and areas of regions given in polar coordinates.

Slopes of parametric curves

When x = x(t) and y = y(t), the slope of the curve is the ratio of their rates, dydx=dy/dtdx/dt\frac{dy}{dx} = \frac{dy/dt}{dx/dt}, wherever dx/dt is not 0.

Example: A slope from rates

Find dy/dx for x = t², y = t³ − 3t at t = 2.

  1. dx/dt = 2t, which is 4, and dy/dt = 3t² − 3, which is 9.
  2. dy/dx = 9 ÷ 4.

Answer2.25

Example: A circle

x = 3 cos t, y = 3 sin t. Where is the tangent horizontal, and where vertical?

  1. dy/dt = 3 cos t is 0 at t = π/2 and 3π/2: horizontal at the top and bottom.
  2. dx/dt = −3 sin t is 0 at t = 0 and π: vertical at the two sides.

AnswerHorizontal at (0, ±3), vertical at (±3, 0)

Arc length of parametric curves

The speed along a path is √((dx/dt)² + (dy/dt)²). Integrating the speed over time gives the length of the path.

L=∫ab(dxdt)2+(dydt)2 dtL = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^{2} + \left(\frac{dy}{dt}\right)^{2}}\,dt

Example: A circle, as a check

Find the length of x = 3 cos t, y = 3 sin t for 0 ≤ t ≤ 2π.

  1. The speed is √(9 sin²t + 9 cos²t), which is 3.
  2. Speed 3 for a time of 2π.

Answer6π, the circumference

Example: A curve

Find the length of x = t², y=23t3y = \frac{2}{3}t^{3} for 0 ≤ t ≤ 1.

  1. Speed: 4t2+4t4=2t1+t2\sqrt{4t^{2} + 4t^{4}} = 2t\sqrt{1 + t^{2}}.
  2. With u = 1 + t², ∫012t1+t2 dt=23(22−1)\int_{0}^{1} 2t\sqrt{1 + t^{2}}\,dt = \frac{2}{3}\left(2\sqrt{2} - 1\right).

AnswerAbout 1.22

Area in polar coordinates

A region swept out by r = f(θ) from θ = α to θ = β is made of thin wedges. A wedge of radius r and angle dθ has area ½r² dθ, like a slice of pie.

A=∫αβ12r2 dθA = \int_{\alpha}^{\beta} \frac{1}{2} r^{2}\,d\theta
The cardioid r = 2(1 + cos θ). Its area is 6π.

Example: A circle, as a check

Find the area inside r = 4.

  1. ∫02π12(16) dθ=16π\int_{0}^{2\pi} \frac{1}{2}(16)\,d\theta = 16\pi.

Answer16π, as πr² says

Example: A cardioid

Find the area inside r = 2(1 + cos θ).

  1. 12∫02π4(1+cos⁡θ)2 dθ\frac{1}{2}\int_{0}^{2\pi} 4(1 + \cos\theta)^{2}\,d\theta.
  2. Over a full turn, 1 + 2 cos θ + cos²θ integrates to 2π + 0 + π, which is 3π.
  3. Times 2 gives 6π.

Answer6π

Try it yourself

Work each one on paper first, then open the answer to check.

  1. Find dy/dx for x = 2t, y = t² at t = 3.

    Show answer

    3

  2. Find dy/dx for x = cos t, y = sin t at t = π/4.

    Show answer

    −1

  3. Where does x = t² − 4t, y = t³ have a vertical tangent?

    Show answer

    At t = 2, the point (−4, 8)

  4. Find the length of x = 3t, y = 4t for 0 ≤ t ≤ 2.

    Show answer

    10

  5. Find the area inside r = 3.

    Show answer

    9π

  6. Find the area inside r = 2 sin θ for 0 ≤ θ ≤ π.

    Show answer

    π

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