find the area of the region inside the circle r = 3sin(teta) and outside the cardioid r = 1 + sin(teta)
First make a sketch of the situation. Then perform an area integration of the difference between the two functions, using polar coordinates. Your sketch should tell you what limits of angle theta integration for use. It looks like it might be 0 degrees to pi.
To find the area of the region inside the circle and outside the cardioid, you can use the concept of double integrals in polar coordinates. Here's how you can approach the problem:
1. First, let's set up the limits of integration. Since we are dealing with polar coordinates, we'll integrate with respect to theta (θ) in the range from 0 to 2π.
2. Let's find the limits for the radial component (r) based on the given circle and cardioid equations. The circle has a radius of r = 3sin(θ). We can set its limits to be from 0 (at the origin) to the maximum value of the circle, which occurs at θ = π/2. The cardioid has a radius of r = 1 + sin(θ). We need to find where the circle and cardioid intersect to determine the lower limit for the cardioid.
3. Set the two equations r = 3sin(θ) and r = 1 + sin(θ) equal to each other to find the intersection points. Simplifying the equation, we get 3sin(θ) = 1 + sin(θ), which yields sin(θ) = 1/2. Solving for θ, we find two values: θ = π/6 and θ = 5π/6. Since we want the area outside the cardioid, we'll take the upper intersection point as the lower limit. Therefore, the limits for r will be from 1 + sin(θ) to 3sin(θ), and the limits for θ will be from π/6 to 2π.
4. Now, we're ready to set up the double integral. The formula to calculate area using double integrals in polar coordinates is:
A = ∫∫ r dr dθ
The limits of integration for θ are from π/6 to 2π, and for r, they are from 1 + sin(θ) to 3sin(θ).
Therefore, the integral becomes:
A = ∫(π/6 to 2π) ∫(1 + sin(θ) to 3sin(θ)) r dr dθ
5. Evaluate the integral to find the area. It can be a bit challenging to solve analytically, so you may consider using numerical methods or software to compute the result.
By following these steps, you should be able to find the area of the region inside the circle and outside the cardioid.