Question

Find the area lying outside r=6sinθ and inside r=3+3sinθ.

Answer 1

To find the area lying outside r=6sinθ and inside r=3+3sinθ, we need to graph the two polar equations and determine the boundaries of the region. The equation r=6sinθ represents a cardioid, while r=3+3sinθ represents a limaçon. The points where the two graphs intersect will give us the boundaries of the desired region.

Step-by-step explanation:

To find the area lying outside r=6sinθ and inside r=3+3sinθ, we need to graph the two polar equations and determine the boundaries of the region. The equation r=6sinθ represents a cardioid, while r=3+3sinθ represents a limaçon. The points where the two graphs intersect will give us the boundaries of the desired region. We can then calculate the area using the formula for the area of a polar region.

The graph of r=6sinθ is a cardioid with a radius of 6 and the graph of r=3+3sinθ is a limaçon with a radius of 6 at its maximum. The points of intersection occur at θ=0 and θ=π.

To calculate the area, we integrate the function 1/2*(6sinθ)^2 - 1/2*(3+3sinθ)^2 with respect to θ from 0 to π. Evaluating this integral gives the area lying outside the cardioid and inside the limaçon.

Answer 2

To find the area lying outside r=6sinθ and inside r=3+3sinθ, set the two equations equal to each other and solve for θ. Then, integrate the area between the curves using the bound α=0 and β=π/2.

Step-by-step explanation:

To find the area lying outside r=6sinθ and inside r=3+3sinθ, we need to understand the region between these two curves. First, we need to find the points of intersection of the two curves. Set them equal to each other:

6sinθ = 3+3sinθ

Subtracting 3sinθ from both sides, we get: 3sinθ = 3

Dividing by 3, we have: sinθ = 1

Since sinθ = 1 only at θ = π/2, we know there is only one intersection point. The region between the two curves is bounded by the angles of θ = 0 and θ = π/2. To find the area, we integrate the area between the curves:

Area = ∫[0,π/2] [(3+3sinθ)^2 - (6sinθ)^2] dθ

Your result will depend on solving the integral.