Using an appropriate algebraic substitution, express the polar-coordinates area

Question

Using an appropriate algebraic substitution, express the polar-coordinates area formula 1/2 [f(theta)]2 d theta so that the angles alpha, beta have unitsof degrees instead of radians Graphs of the functions y = f(x), y = g(x), y = h(x), and y = j(x) in Cartesian coordinates are displayed below. Translate the information provided in these graphs to sketch the polar curves r = f(theta),r = g(theta),r = h(theta), and r = j(theta). Find the area of the "inside loop" of the limacon r = 2 + 4sin(theta). You may want sketch y = 2 + 4sin(x) first. If f is an even function, show that the graph of r = f(theta) is symmetric about the x-axis. Will this ever give rise to a function in the Cartesian sense? Explain.

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