Calculating Polar Areas

Key Questions

• How do you find the area of the region bounded by the polar curve #r=2+cos(2theta)# ?

  The area inside a polar curve is approximately the sum of lots of skinny wedges that start at the origin and go out to the curve, as long as there are no self-intersections for your polar curve.

  Each wedge or slice or sector is like a triangle with height #r# and base #r# #dθ#, so the area of each element is
  #dA = 1/2 b h = 1/2 r (r dθ) = 1/2 r^2 dθ.#

  So add them up as an integral going around from θ=0 to θ=2π, and using a double angle formula, we get:

  #A = 1/2 int_0 ^(2π)(2 + cos(2θ))^2 dθ#

  #A = 1/2 int_0 ^(2π) [4 + 4 cos(2θ) + cos^2(2θ)] dθ#

  #A = 1/2 int_0 ^(2π) [4 + 4 cos(2θ) + (1 + cos(4θ))/2] dθ.#

  Now do the integral(s) by subbing u = 2θ and then u = 4θ, and remember to change limits for the "new u." I'll let you evaluate those to get practice integrating! Remember our motto,

  "Struggling a bit makes you stronger." \dansmath/

  dansmath · 3 · Oct 6 2014
• How do you find the area of the region bounded by the polar curves #r=3+2cos(theta)# and #r=3+2sin(theta)# ?

  Let us look at the region bounded by the polar curves, which looks like:

  

  Red: #y=3+2cos theta#
  Blue: #y=3+2sin theta#
  Green: #y=x#

  Using the symmetry, we will try to find the area of the region bounded by the red curve and the green line then double it.

  #A=2int_{pi/4}^{{5pi}/4}\int_0^{3+2cos theta}rdrd theta#

  #=2int_{pi/4}^{{5pi}/4}[r^2/2]_0^{3+2cos theta} d theta#

  #=int_{pi/4}^{{5pi}/4}(9+12cos theta+4cos^2theta)d theta#

  by #cos^2theta=1/2(1+cos2theta)#,

  #=int_{pi/4}^{{5pi}/4}(11+12cos theta+2cos2theta)d theta#

  #=[11theta+12sin theta+sin2theta]_{pi/4}^{{5pi}/4}#

  #={55pi}/4-6sqrt{2}+1-({11pi}/4+6sqrt{2}+1)#

  #=11pi-12sqrt{2]#

  Hence, the area of the region is #11pi-12sqrt{2}#.

  ---

  I hope that this was helpful.

  Wataru · 3 · Nov 8 2014
• How do you find areas bounded by polar curves using calculus?

  If the region is bounded by a polar curve #r=r(theta)# from #theta=theta_1# to #theta_2#, then its area #A# can be found by the double-integral

  #A=int_{theta_1}^{theta_2}int_0^{r(theta)}rdrd theta#.

  ---

  I hope that this was helpful.

  Wataru · · Oct 18 2014

• How do you find the area of the region bounded by the polar curve #r=3cos(theta)# ?
• How do you find the area of the region bounded by the polar curve #r=3(1+cos(theta))# ?
• How do you find the area of the region bounded by the polar curve #r=2-sin(theta)# ?
• How do you find the area of the region bounded by the polar curve #r^2=4cos(2theta)# ?
• How do you find the area of the region bounded by the polar curve #r=2+cos(2theta)# ?
• How do you find the area of the region bounded by the polar curves #r=sqrt(3)cos(theta)# and #r=sin(theta)# ?
• How do you find the area of the region bounded by the polar curves #r=1+cos(theta)# and #r=1-cos(theta)# ?
• How do you find the area of the region bounded by the polar curves #r=cos(2theta)# and #r=sin(2theta)# ?
• How do you find the area of the region bounded by the polar curves #r^2=cos(2theta)# and #r^2=sin(2theta)# ?
• How do you find the area of the region bounded by the polar curves #r=3+2cos(theta)# and #r=3+2sin(theta)# ?
• If a sprinkler distributes water in a circular pattern, supplying water to a depth of #e^-r# feet per hour at a distance of r feet from the sprinkler, what is the total amount of water supplied per hour inside of a circle of radius 11?
• How do you use the polar coordinates to find the volume of a sphere of radius 10?
• How do you evaluate the integral #sin(x^2+y^2)dr# where r is the region #9<= x^2 + y^2 <= 64# in polar form?
• How do you find the area between #r=1+cos(theta#) and #r=1-cos(theta)#?
• How do you use polar coordinates to evaluate the integral which gives the area that lies in the first quadrant between the circles #x^2+y^2=36# and #x^2-6x+y^2=0#?
• What is the area enclosed by #r=theta # for #theta in [0,pi]#?
• What is the area enclosed by #r=theta^2-2sintheta # for #theta in [pi/4,pi]#?
• What is the area enclosed by #r=thetacostheta-2sin(theta/2-pi) # for #theta in [pi/4,pi]#?
• What is the area enclosed by #r=theta^2cos(theta+pi/4)-sin(2theta-pi/12) # for #theta in [pi/12,pi]#?
• What is the area enclosed by #r=cos(theta-(7pi)/4)+sin(-theta-(9pi)/12) # between #theta in [pi/12,(3pi)/2]#?
• What is the area enclosed by #r=cos(4theta-(7pi)/4)+sin(theta+(pi)/8) # between #theta in [pi/3,(5pi)/3]#?
• What is the area enclosed by #r=-cos(theta-(7pi)/4) # between #theta in [4pi/3,(5pi)/3]#?
• What is the area enclosed by #r=-thetasin(-16theta^2+(7pi)/12) # between #theta in [0,(pi)/4]#?
• What is the area enclosed by #r=2sin(4theta+(11pi)/12) # between #theta in [pi/8,(pi)/4]#?
• What is the area enclosed by #r=sin(5theta-(13pi)/12) # between #theta in [pi/8,(pi)/4]#?
• What is the area enclosed by #r=2sin(theta+(7pi)/4) # between #theta in [pi/8,(pi)/4]#?
• What is the area enclosed by #r=-sin(3theta-(7pi)/4) # between #theta in [pi/8,(pi)/4]#?
• What is the area enclosed by #r=8sin(3theta-(2pi)/4) +4theta# between #theta in [pi/8,(pi)/4]#?
• What is the area enclosed by #r=-3sin(2theta-(2pi)/4) -theta# between #theta in [pi/8,(3pi)/4]#?
• What is the area enclosed by #r=5sin(-6theta-(5pi)/8) -2theta# between #theta in [pi/8,(3pi)/4]#?
• What is the area enclosed by #r=-sin(theta+(11pi)/8) -theta/4# between #theta in [0,(pi)/2]#?
• What is the area enclosed by #r=-sin(theta+(15pi)/8) -theta# between #theta in [0,(pi)/2]#?
• What is the area enclosed by #r=costheta-sintheta/2 -3theta# between #theta in [0,(pi)/2]#?
• What is the area enclosed by #r=2cos((2theta)/3+(5pi)/3)+4sin(theta/2+pi/4) -theta# between #theta in [0,pi]#?
• What is the area enclosed by #r=7cos((theta)/12-(3pi)/2)+2sin((2theta)/3+(2pi)/3) +theta/3# between #theta in [0,pi]#?
• What is the area enclosed by #r=-2cos((11theta)/12+(3pi)/4)+sin((5theta)/4+(5pi)/4) +theta/3# between #theta in [0,pi]#?
• What is the area enclosed by #r=cos((2theta)/3-(pi)/8)+sin((7theta)/8+(pi)/4) # between #theta in [0,pi]#?
• What is the area enclosed by #r=2cos((5theta)/3-(13pi)/8)-3sin((5theta)/8+(pi)/4) # between #theta in [0,pi]#?
• What is the area enclosed by #r=-9cos((5theta)/3-(pi)/8)+3sin((theta)/2+(3pi)/4) # between #theta in [0,pi/2]#?
• What is the area enclosed by #r=5cos^3((5theta)/6-(pi)/2)+theta^2/2 # between #theta in [0,pi/2]#?
• What is the area enclosed by #r=sintheta/theta-theta^3-theta # between #theta in [pi/12,pi/3]#?
• What is the area under the polar curve #f(theta) = theta # over #[0,2pi]#?
• What is the area under the polar curve #f(theta) = thetasin((3theta)/4 )-cos^3((5theta)/12-pi/2) # over #[0,2pi]#?
• What is the area under the polar curve #f(theta) = theta^2sin((5theta)/2 )-cos((2theta)/3+pi/2) # over #[pi/6,(3pi)/2]#?
• What is the area under the polar curve #f(theta) = theta-thetasin((7theta)/8 )-cos((5theta)/3+pi/3) # over #[pi/6,(3pi)/2]#?
• What is the area under the polar curve #f(theta) = thetasin(-theta )+2cot((7theta)/8) # over #[pi/4,(5pi)/6]#?
• What is the area under the polar curve #f(theta) = theta^2-thetasin(2theta-pi/4 ) +cos(3theta-(5pi)/4)# over #[pi/8,pi/2]#?
• What is the area under the polar curve #f(theta) = theta^2-thetasin(7theta-pi/6 ) +cos(2theta-(5pi)/4)# over #[pi/8,pi/2]#?
• What is the area under the polar curve #f(theta) = theta^2-thetasin(6theta-pi/12 ) +cos(12theta-(5pi)/3)# over #[pi/8,pi/6]#?
• What is the area under the polar curve #f(theta) = 3theta^2+thetasin(4theta-(5pi)/12 ) +cos(2theta-(pi)/3)# over #[pi/8,pi/6]#?
• Show that # int_0^h int_0^x sqrt(x^2+y^2) dy dx = h^3/6 (sqrt(2) + ln( sqrt(2) + 1) ) #?
• How do you find the area of one petal of #r=2cos3theta#?
• How do you find the area of one petal of #r=6sin2theta#?
• How do you find the area of one petal of #r=cos2theta#?
• How do you find the area of one petal of #r=cos5theta#?
• How do you find the area inner loop of #r=4-6sintheta#?
• How do you find the area between the loop of #r=1+2costheta#?
• How do you find the area between the loops of #r=2(1+2sintheta)#?
• How do you find the points of intersection of #r=3(1+sintheta), r=3(1-sintheta)#?
• How do you find the points of intersection of #r=1+costheta, r=1-sintheta#?
• How do you find the points of intersection of #r=2-3costheta, r=costheta#?
• How do you find the points of intersection of #r=4-5sintheta, r=3sintheta#?
• How do you find the points of intersection of #r=1+costheta, r=3costheta#?
• How do you find the points of intersection of #r=theta/2, r=2#?
• How do you find the points of intersection of #theta=pi/4, r=2#?
• How do you find the points of intersection of #r=3+sintheta, r=2csctheta#?
• How do you find the area of the common interior of #r=3-2sintheta, r=-3+2sintheta#?
• How do you find the area of the common interior of #r=4sintheta, r=2#?
• How do you find the area the region of the common interior of #r=a(1+costheta), r=asintheta#?
• Question #4e7fa
• Given the parametric equations #x=acos theta# and #y=b sin theta#. What is the bounded area?
• Find the area bounded by the inside of the polar curve # r=1+cos 2theta # and outside the polar curve # r=1 #?
• Find the area of a loop of the curve #r=a sin3theta#?
• Question #f1f96
• What is the area inside the polar curve #r=1#, but outside the polar curve #r=2costheta#?
• What is the area bounded by the the inside of polar curve #1+cos theta# and outside the polar curve #r(1+cos theta)=1#?
• Calculating areas bounded by polar curves looks extremely difficult. Do Americans really need to integrate such a complex expressions without a calculator?
• Find the area bounded by the polar curves? #r=4+4cos theta# and #r=6#
• Using integrals, find the area of the circle #x^2 + y^2 = 1# ?
• Find the area of a single loop in curve #r=\sin(6\theta)#?
• Find the area of the region inside these curves?