Generalization of Determining Area and Arc Length of Cosine Function in Polar Coordinate

Abstract: Polar coordinate is formed by the radius and the rotation angle. The radius indicates the distance of a point from the pole, while the rotation angle is the angle from which starting point moves to ending point relative to the axis-x. Because polar coordinate is affected by two variables, polar coordinate is defined on a plane. This research aims to find a general formula for determining area and length of a curve. The curve equation is a cosine function, which is circular and cardioid. The graph of circle is affected by constant and positive or negative sign on the cosine function. The constant determines the size of the graph, while positive or negative sign indicates position of the graph in the quadrant. Similarly, the graph of cardioid is affected by constant and positive or negative sign on the cosine function. The constant determines the size of the graph, while positive or negative sign indicates the direction of the cardioid. The area of circle is directly proportional to the square of constant. The arc length of circle is directly proportional to the constant. The area of cardioid is directly proportional to the square of constant. The arc length of cardioid is equal to eight times the constant. The general formula for the area and the arc length of the curve is used to predict their values. Simulations using several constant values are performed to practice the general formula for area and arc length of the circle and the cardioid.