Questions: Find the area shared by the circle (r2=4) and the cardioid (r1=4(1-cos theta)). The area shared by the circle and the cardioid is (square) (Type an exact answer, using (pi) as needed.)

Find the area shared by the circle (r2=4) and the cardioid (r1=4(1-cos theta)).

The area shared by the circle and the cardioid is (square)
(Type an exact answer, using (pi) as needed.)

Transcript text: Find the area shared by the circle $r_{2}=4$ and the cardioid $r_{1}=4(1-\cos \theta)$. The area shared by the circle and the cardioid is $\square$ (Type an exact answer, using $\pi$ as needed.)

Solution

Solution Steps

To find the area shared by the circle $ r_2 = 4 $ and the cardioid $ r_1 = 4(1-\cos \theta) $, we need to determine the points of intersection and then integrate the area between these curves over the appropriate range of $\theta$. The intersection points can be found by setting the equations equal to each other. Once the intersection points are determined, the area can be calculated by integrating the difference of the squares of the radii of the two curves over the interval defined by these points.

Step 1: Define the Equations

We have two polar equations: the circle $ r = 4 $ and the cardioid $ r = 4(1 - \cos \theta) $.

Step 2: Find Points of Intersection

To find the points of intersection, we set the equations equal to each other: \[ 4 = 4(1 - \cos \theta) \] Solving for $\theta$, we get: \[ 1 = 1 - \cos \theta \implies \cos \theta = 0 \] The solutions for $\theta$ are $\theta = \frac{\pi}{2}$ and $\theta = \frac{3\pi}{2}$.

Step 3: Calculate the Shared Area

The area shared by the circle and the cardioid is given by the integral: \[ \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \frac{1}{2} \left( (4(1 - \cos \theta))^2 - 4^2 \right) \, d\theta \] Evaluating this integral, we find: \[ \text{Area} = 4\pi + 32 \]