Final answer:
To find the area of the region outside r=10+10 sin θ, but inside r=30 sin θ, we need to find the points where these two curves intersect and calculate the area of the sector. Then, subtract the area of the sector from the area of the circle defined by the second curve to find the desired area.
Step-by-step explanation:
To find the area of the region outside r=10+10 sin θ, but inside r=30 sin θ, we need to find the points where these two curves intersect. By setting the equations equal to each other and solving for θ, we can find the points where the curves intersect. Then, we can calculate the area of the sector using the formula A = 0.5r²θ, where r is the radius and θ is the angle between the two curves at the point of intersection. Finally, we can subtract the area of the sector from the area of the circle defined by the second curve to find the area of the region outside the first curve but inside the second curve.