Find the area of the region enclosed by the inner loop of the curve.
r = 5 + 10sin(θ)
Any help would be greatly appreciated!!!
To find the area of the region enclosed by the inner loop of the curve described by the polar equation r = 5 + 10sin(θ), you can follow these steps:
1. Sketch the curve: Start by sketching the polar curve on a polar coordinate system. This will help you visualize the shape of the curve and determine which part encloses the area you are interested in.
2. Determine the limits of integration: To find the area enclosed by the inner loop, you need to determine the limits of integration for θ. You can do this by finding the values of θ where the curve intersects itself and encloses the desired region. In this case, the inner loop completes when sin(θ) is equal to -1, which occurs when θ = π.
3. Set up the integral: The area of the region enclosed by the inner loop can be calculated by integrating the function 1/2 * r^2 with respect to θ, where r is the polar equation. In this case, the integral would be ∫(1/2 * (5 + 10sin(θ))^2)dθ, with the limits of integration from 0 to π.
4. Calculate the integral: Evaluate the integral using appropriate integration techniques, such as u-substitution or trigonometric identities. This will give you the area of the region enclosed by the inner loop of the curve.
By following these steps, you will be able to find the area of the region enclosed by the inner loop of the given polar curve.