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Area enclosed by a polar curveAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

  • Area enclosed by a polar curve between \theta=\alpha and \theta=\beta?
  • In what units must \theta be for the polar area formula?
  • Where does the formula \frac12\int r^2\,d\theta come from?
  • Identity for \cos^2\theta?
  • Identity for \sin^2\theta?
  • Limits for the whole cardioid r=a(1+\cos\theta)?
  • Limits for the circle r=a\cos\theta traced once?
  • Limits for one loop of r=a\sin2\theta?
  • Area between an outer curve r1 and an inner curve r2?
  • How do you find the limits for the area between two curves?
  • Area enclosed by r=a using the formula?
  • Area of r=1+\cos\theta over 0\le\theta\le2\pi?

Exam questions on Area enclosed by a polar curve

  1. A spiral has polar equation r=2θr=2\theta for 0≤θ≤π0\le\theta\le\pi.
    Find the exact area of the region bounded by the spiral and the half-lines θ=π2\theta=\frac{\pi}{2} and θ=π\theta=\pi.2 marks
  2. A curve CC has polar equation r=3+2cos⁡θr=3+2\cos\theta for 0≤θ≤2π0\le\theta\le2\pi.
    Hence find the area enclosed by CC.2 marks
  3. A curve CC has polar equation r=4sin⁡2θr=4\sin2\theta for 0≤θ≤π20\le\theta\le\frac{\pi}{2}, forming one loop from the pole.
    Show that the area enclosed by the loop is 2π2\pi.3 marks
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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).