Question

Find the area inside the larger loop and outside the smaller loop of the limaçon r = ¹/₂ cos(θ).

Answer 1

To find the area inside the larger loop and outside the smaller loop of the limaçon r = ¹/₂ cos(θ), follow these steps: determine the boundaries of the loops, find the area inside the larger loop using integration, find the area inside the smaller loop using integration, and subtract the area of the smaller loop from the area of the larger loop.

Step-by-step explanation:

To find the area inside the larger loop and outside the smaller loop of the limaçon r = ¹/₂ cos(θ), we need to identify the boundaries of the loops. The larger loop is determined by the equation r = ¹/₂ cos(θ), and the smaller loop is determined by r = 0. The area between the two loops can be found by subtracting the area of the smaller loop from the area of the larger loop. Here are the steps to find the area:

1. Determine the boundaries of the loops by solving the equations r = ¹/₂ cos(θ) and r = 0.
2. Find the area inside the larger loop by integrating the equation of the limaçon within the boundaries of the loop.
3. Find the area inside the smaller loop by integrating the equation of r = 0 within the boundaries of the loop.
4. Subtract the area of the smaller loop from the area of the larger loop to get the final area between the two loops.