Final answer:
To find the area of the curve defined by R=3−Cos3θ that is above the x-axis, we need to convert the equation to Cartesian coordinates, find the limits of integration, and evaluate the definite integral. The approximate area under the curve is 59.690 units.
Step-by-step explanation:
To find the area of the curve defined by R=3−Cos3θ that is above the x-axis, we need to find the area enclosed by the curve within one complete revolution. We can do this by integrating the equation of the curve and then using the definite integral to find the area. Since the curve is defined in polar coordinates, we need to convert the equation to Cartesian coordinates to find the limits of integration.
Converting the equation to Cartesian coordinates, we have x = r*cos(θ) and y = r*sin(θ). Substituting these values into the equation, we get R = 3 - cos(3*atan2(y, x)). The limits of integration will then be from the angle where the curve intersects the x-axis to the angle where the curve completes one revolution.
Once we have the limits of integration, we can evaluate the definite integral to find the area under the curve. Using numerical methods or a graphing calculator, the area is approximately 59.690 units.