Verified answer

the area in plane polar coordinates is

A = double integral (r dr dtheta) = 1/2 integral(r^2 d(theta))

find the limits of integration of the inner loop by setting r(theta) =0

and find that r=0 when theta = pi/3 or - pi/3

(note that this function is even, so you can use symmetry to make your integration even easier)

it should help you to plot a few points, when theta = 0, r = -1, so the inner loop starts with theta equal zero and reaches the origin at theta = pi/3, thus the area of the inner loop is

2 x 1/2 Integral[(1- 2 cos(theta)^2 d(theta)] from theta =0 to theta = pi/3

use the same reasoning to set the limits for the outer loop, using symmetry again to make your life a little easier

(for the inner loop, I get area = pi - 3Sqrt[3]/2, for the outer loop 2 pi + 3Sqrt[3]/2)

R = cos? is extremely interior r = a million+cos? discipline (r = a million+cos?) = a million/2 of ? [0 to 2?] r² d? = a million/2 of ? [0 to 2?] (a million + cos?)² d? = a million/2 ? [0 to 2?] (a million + 2cos? + cos²?) d? = a million/2 of ? [0 to 2?] (a million + 2cos? + a million/2 + a million/2 of cos(2?)) d? = a million/2 ? [0 to 2?] (3/2 + 2cos? + a million/2 cos(2?)) d? = a million/2 (3/2 ? + 2sin? + a million/4 sin(2?)) a million/2 of [(3/2 (2?) + 0 + 0) ? (0 + 0 + 0)] = three?/2 field (r = cos?) = a million/2 ? [0 to ?] r² d? = a million/2 of ? [0 to ?] cos²? d? = a million/2 ? [0 to ?] (a million/2 + a million/2 cos(2?)) d? = a million/2 of (a million/2 of ? + a million/4 [0 to ?] = a million/2 [(a million/2 (?) + 0 + 0) ? ( 0 + 0)] = ?/4 A = area (r = a million+cos?) ? discipline (r = cos?) A = 3?/2 ? ?/4 A = 5?/4