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?