Final answer:
The area of the region bounded by the curve r = 7 and lying in the sector between θ = 2 and θ = 2π is found using the formula for a sector's area, resulting in 24.5π - 49 square units.
Step-by-step explanation:
To find the area of the region bounded by the curve r = 7 that lies in the specified sector between θ = 2 and θ = 2π, we employ the formula for the area of a sector of a circle, A = ½ r²Δθ, where r is the radius and Δθ is the change in angle.
In this case, the radius (r) is constant at 7. Since the sector starts at θ = 2 radians and ends at θ = 2π radians, the change in angle (Δθ) is the difference between these two, which is 2π - 2.
The area (A) can then be calculated using the equation:
A = ½ × 7² × (2π - 2)
After solving, we get A = ½ × 49 × (2π - 2), which equals 24.5π - 49 square units.