find the area of the region described:
shared by the cardiods r = 2(1+cos theta) and r = 2(1-cos theta)
To find the area of the region shared by the two cardioids, you need to integrate the area element over the region. First, let's determine the limits of integration and the expression for the area element.
The two cardioid equations are:
1. r = 2(1 + cos(theta))
2. r = 2(1 - cos(theta))
The cardioid given by the equation 1 is traced out when theta varies from 0 to 2π. Similarly, the cardioid given by equation 2 is traced out when theta varies from 0 to 2π.
To find the area, we need to evaluate the following double integral in polar coordinates:
A = ∬ D r dr dθ
Where D represents the region enclosed by the two cardioids.
First, we'll express the area element dA in terms of dr and dθ. In polar coordinates, the differential area element is given by r dr dθ.
Now, let's solve for the limits of integration.
1. Determine the angle range for the shared region:
Since both cardioids trace out a complete circle between 0 and 2π, the angle range for the shared region is from 0 to 2π.
2. Determine the radius range for the shared region:
The shared region lies between the two cardioids. To find the radius range, we set the two cardioid equations equal to each other and solve for r:
2(1 + cos(theta)) = 2(1 - cos(theta))
2 + 2cos(theta) = 2 - 2cos(theta)
4cos(theta) = -2
Divide both sides by 4:
cos(theta) = -1/2
Using the unit circle, we find two values for theta where cos(theta) = -1/2: theta = 2π/3 and theta = 4π/3.
Thus, the radius range for the shared region is from r = 2(1 - cos(theta)) to r = 2(1 + cos(theta)), or in other words, from r = 1 to r = 3.
Now, we have determined the limits of integration:
θ varies from 0 to 2π,
r varies from 1 to 3.
With these limits, we can now evaluate the area integral as follows:
A = ∫[0 to 2π] ∫[1 to 3] (r dr dθ)
Evaluating this double integral will give you the area shared by the two cardioids.