<prompt> Mviringo wa polar r 1 cos theta huzungushwa mara moja karibu na uhakika na viwianishi vya polar 2,0. Ni eneo gani la eneo ambalo linafagia?  <response>: Tunapanga r 1 cos theta. Ikiwa tutaizungusha karibu na uhakika 2,0, basi mduara unafagia mduara wa radius R, ambapo R ndio umbali wa juu kati ya hatua kwenye curve na uhakika 2,0.asyukubwa wa kitengo 1 cm;jozi moo t halisi halisi r 1 cos t; kurudi rcost, rsint;njia foo moo0;t halisi;kwa t 0; t 2pi 0.1; t t 0.1 foo foo--moot;kujazaMduara2,0,4sqrt3,kijivu0.9,kijivu0.9;kuchora foo;nukta 2,0, nyekundu;lebo2,0, 2,0, E;asyKwa curve r 1 cos theta,anzaalignx r cos theta 1 cos theta cos theta, y r sin theta 1 cos theta sin theta,endalignkwa hivyo ikiwa d ni umbali kati ya x,y na 2,0, basianzaalignd2 1 cos theta cos theta - 22 1 cos theta sin theta2 cos2 theta cos theta - 22 1 cos theta2 sin2 theta cos2 theta cos theta - 22 1 cos theta2 1 - cos2 theta cos4 theta 2 cos3 theta - 3 cos2 theta - 4 cos theta 4 -cos4 theta - 2 cos3 theta 2 cos theta 1 -3 cos2 theta - 2 cos theta 5 -3 kushoto cos theta frac13 kulia2 frac163.endalignThamani ya juu zaidi ya d2 basi ni frac163, ambayo hutokea wakati cos theta -frac13.Kwa hivyo, eneo ambalo curve inafagia ni frac16 pi3. 