When recording live performances, sound engineers often use a microphone with a cardioids pickup pattern because it suppresses noise from the audience. suppose the microphone is placed 5 m from the front on the stage and the boundary of the optimal pickup region is given by the given cardioids, where r is measured in meters and the microphone is at the pole. the musicians want to know the area they will have on stage within the optimal pickup range of the microphone. the answer should be in two decimal places. r = 10 + 10sin(θ)

R = 4 + 4sinθ 
y = rsinθ > 3 when π/6 < θ < 5π/6 

So to find the shaded area we need to integrate 
dA = rdr dθ 
between the limits 
r = 3/sinθ to 4 + 4sinθ 
and θ = π/6 to 5π/6 

You need to integrate r first, and you'll get 
∫ rdrdθ = ∫ [r^2 /2] dθ 
= ∫ [(4+4sinθ)^2 - (3cscθ)^2] dθ 
= ∫ [16 + 32sinθ + 16sin^2 θ - 9csc^2 θ] dθ 
= [24θ - 32cosθ + 9cotθ - 4sin(2θ)] 

Apply the limits to get 
16π + 18√3 

Acetic acid  /əˈsiːtɪk/, systematically named  ethanoic acid  /ˌɛθəˈnoʊɪk/, is a colourless liquid  organic compound  with the  chemical formula  ch3cooh (also written as ch3co2h or c2h4o2). when undiluted, it is sometimes called  glacial acetic acid.  vinegar  is no less than 4% acetic acid by volume, making acetic acid the main component of vinegar apart from water. acetic acid has a distinctive sour taste and pungent smell. in addition to household vinegar, it is mainly produced as a precursor to  polyvinyl acetate  and  cellulose acetate. it is classified as a  weak acid  since it only partially  dissociates  in solution, but concentrated acetic acid is corrosive and can attack the skin.acetic acidnamespreferred iupac nameacetic acidsystematic iupac nameethanoic acidother namesvinegar (when dilute); hydrogen acetate; methanecarboxylic acid[1][2]identifierscas number64-19-7  3d model (jsmol)interactive image3dmetb00009 reference506007chebichebi: 15366  chemblchembl539  chemspider171  drugbankdb03166  echa infocard100.000.528ec number200-580-7e numbere260  (preservatives)gmelin reference1380iuphar/bps1058keggd00010  meshacetic+acidpubchem  cid176rtecs numberaf1225000uniiq40q9n063p  un number2789inchiinchi=1s/c2h4o2/c1-2(3)4/h1h3,(h,3,4)  key:   qtbsbxvteameqo-uhfffaoysa-n  smilescc(o)=o