How do you find the region inside cardioid ##r=1+cos(theta)## and outside the circle ##r=3cos(theta)##?

It is ##pi/4##

Find the intersection points of the curves hence we have that
##3cosθ=1+cosθ=>cosθ=1/2=>θ=+-pi/3##
The saded area is
(cardiod area from pi/3 to pi)-(cricle area from pi/3 to pi/2)
The cardiod area is
##int_(pi/3)^(pi) 1/2*(1+cosθ)^2dθ=pi/2-9/6*sqrt3##
and the circle area is
##int_(pi/3)^(pi/2) 1/2*(3*cosθ)^2dθ=(3pi/8)-9/16*sqrt3##
Hence the shaded area is ##pi/8##
The total amount is ##2pi/8=pi/4##
A graph for the curves is

Posted in Uncategorized

Leave a Reply

Your email address will not be published. Required fields are marked *

You may use these HTML tags and attributes:

<a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <s> <strike> <strong>