Final answer:
To find the area of the region inside the circle r=20cosθ and to the right of the vertical line r=5secθ, we need to determine the bounds of integration. The area can be found using the formula for the area enclosed by a polar curve. The bounds of integration are θ = cos⁻¹(1/4) to θ = cos⁻¹(4).
Step-by-step explanation:
To find the area of the region inside the circle r=20cosθ and to the right of the vertical line r=5secθ, we need to determine the bounds of integration. The polar curves intersect when 20cosθ = 5secθ. Simplifying this equation gives us 4cosθ = secθ. To find the bounds of integration, we solve for θ by isolating cosθ in terms of secθ. We get cosθ = 1/4 and secθ = 4. Therefore, the bounds of integration are θ = cos⁻¹(1/4) to θ = cos⁻¹(4).
The area of the region can be found using the formula for the area enclosed by a polar curve, which is given by A = 1/2 ∫[θ1,θ2] r² dθ. In this case, r = 20cosθ. Substituting this into the formula and integrating, we get A = 1/2 ∫[cos⁻¹(1/4), cos⁻¹(4)] (20cosθ)² dθ. Simplifying the integral and evaluating it gives the area of the region inside the circle r=20cosθ and to the right of the vertical line r=5secθ.