Generalized Area and Arc Length of Sine Function in Polar Coordinate

Source abstract

Polar coordinate is formed by the radius and rotation angle. The radius indicates distance of a point from pole, while the rotation angle is 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 sine function, which is circular and cardioid. The graph of circle is affected by constant and positive or negative sign on the sine 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 sine 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 arc length of the curve is used as a predictive tool.