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How do you prove that the total area of a four-petaled rose r=sin2x is equal to one-half the area of the circumscribed circle?

  • calculus -

    Use a polar coordinate integration to get the area inside the rose. The circumscribed circle area is, of course, pi, since rmax = 1

    For the rose, the area is
    S (1/2)r^2 dx
    (for x integrated from 0 to 2 pi
    = S (1/2)sin^2x dx
    "S" denotes an integral sign.
    =(1/2)[(x/2) -(1/4)(sin2x)]@x=2pi -
    (1/2)[(x/2) -(1/4)(sin2x)]@x=0
    = (1/2)*pi


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