Final answer:
To determine the area of the inner loop of the polar equation r = 3 - 6 cos θ, one must first use a graphing utility to visualize the graph, then apply the formula for the area in polar coordinates by evaluating the definite integral of (1/2)*r^2 with respect to θ over the relevant interval.
Step-by-step explanation:
The student is asked to graph a polar equation and find the area of the inner loop of the graph of r = 3 - 6 cos θ. To graph this polar equation, typically a graphing utility or software would be used, like a graphing calculator or a computer program that can handle polar coordinates. To find the area of the inner loop analytically, the student would calculate the definite integral of r2/2 with respect to θ, from θ1 to θ2, where θ1 and θ2 are the angles corresponding to the intersection points of the inner loop.
Remember that the complete area formula for a segment in polar coordinates is A = ½ ∫ r2 dθ. By evaluating this integral, the student can obtain the exact area of the inner loop of the given polar equation.