Quadrature and Rectification
In this chapter, we discuss the processes to find area of a given bounded region of a curve and to find length of an arc of a curve between two given points. Of course, these concepts are nothing but applications of definite integral.
The process of finding the area of a bounded region of a curve is called quadrature.
9.1.1 Area of a Curve Given by the Cartesian Equation
Let AB be the arc of the curve y = f(x), between two ordinates x = a and x = b represented by AC and BD respectively.
Let P(x, y) and Q(x+δx, y+δy) be two neighboring points on the given curve. Draw PR and QS perpendiculars ...