Area of Polar function?

Question

Find the area of ONE loop of the graph of the polar function

r = 4cos3Ө

Area = 1/2 INT Ө1 --> Ө2 r^2 dӨ

PLEASE SHOW WORK! Including what the integral begins and goes until.

I can't figure out the limits of the integral... I have no idea where to start and stop to get just ONE loop!

I need the limits for ONE loop. In terms of theta.

for example like pi/2 to pi or something. but that's probably not right.

Answer 1

The polar eq has three loops (3x)
cos 3x = 0 3x = 90 x = 30

so the limit is between +/- 30

Answer 2

the properly-ordinary eqn of a circle is : x^2 + y^2 = r^2 so y = +/- sqrt(r^2 - x^2) the area of the completed circle is two * section below y = sqrt(r^2 - x^2) [from -r to r] => A = 2 * ? (r²-x²)¹/² dx enable x = r cos? => x^2 = r^2 cos^2? , and dx = - r sin? d? so A = -2 * ? sqrt(r^2 - r^2 cos^2?) (r sin? d?) [ from ? to 0 ] ........= -2 * ? r^2sin^2? d? [ from ? to 0 ] ........= -2r^2 ? [a million - cos2?]/2 d? [ from ? to 0 ] ........= - r^2 [ ? - a million/2 sin2?] [ from ? to 0 ] ........= - r^2 (0 - ?) + r^2/2 (sin0 - sin2?) ........= r^2? + 0 ........= r^2 ? <<<<<<<<<<<<<<<<<<<<<<<<<<