Final answer:
To find the length of the entire perimeter, calculate the arc length of the inside curve using integration and find the circumference of the outside circle using the formula C = 2πr.
Step-by-step explanation:
The length of the entire perimeter of the region can be found by calculating the arc length of the inside curve and the circumference of the outside circle.
First, let's find the arc length of the inside curve. The equation for the inside curve is r = 14sinθ. To find the arc length, we need to integrate the square root of the sum of the squares of the derivative of r with respect to θ and the derivative of θ with respect to t, multiplied by dt.
Next, let's find the circumference of the outside circle. The equation for the outside circle is r = 2. To find the circumference, we can use the formula C = 2πr.