Question

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?

Answer 1

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

qed