Question

Use a double integral to find the area of the region. The region inside the cardioid r=1+cos(θ) and outside the circle r=3cos(θ)

Answer 1

To find the area of the region inside the cardioid r=1+cos(θ) and outside the circle r=3cos(θ), you can use a double integral. The limits of integration for theta are from 0 to 2π. The region for r is between 3cos(θ) and 1+cos(θ).

Step-by-step explanation:

In order to find the area of the region inside the cardioid r=1+cos(θ) and outside the circle r=3cos(θ), we can use a double integral.

The first step is to find the limits of integration for theta, which is from 0 to 2π.

The second step is to integrate with respect to r.

Since the region is between the cardioid and the circle, the limits of integration for r are from 3cos(θ) to 1+cos(θ).

Then, we integrate the function rdrdθ over these limits to find the area.

Answer 2

To find the area of the region inside the cardioid r=1+cos(θ) and outside the circle r=3cos(θ), set up a double integral with appropriate limits of integration.

Step-by-step explanation:

To find the area of the region inside the cardioid r = 1 + cos(θ) and outside the circle r = 3cos(θ), we can set up a double integral with the appropriate limits of integration.

The limits of integration for the angle θ are from 0 to 2π, while the limits of integration for the radius r are from the circle r = 3cos(θ) to the cardioid r = 1 + cos(θ).

The double integral is then:

∬ dA = ∫02π ∫3cos(θ)1+cos(θ) r dr dθ

After evaluating this double integral, you will find the area of the region inside the cardioid and outside the circle.