9.8 Evaluation of a Bounded Plane Area by Integration

. Evaluation of a Bounded Plane Area by Integration In the beginning of this chapter, we have already introduced definite integral by a geometrical approach. In that approach, we have noted that, whenever the integrand of the definite integral is non-negative, the definite integral yields the geometrical area. In the present section, we apply the approach for finding areas of plane regions bounded by plane curves. . . Area of the region bounded by a curve, x – axis and the lines x = a and x = b . Case (i) Let y ( ), be the equation of the portion of the continuous curve that lies above the x − axis (that is, the portion lies either in the first quadrant or in the second quadrant) between the lines x and x . See Fig. . . Then, y ≥ for every point of the portion of the curve. Consider the region bounded by the curve, x − axis, the ordinates x and x . It is important to note that y does not change its sign in the region. Then, the area A of the region is found as follows: Fig. . ∆ x O of - - Applications of Integration Viewing in the positive direction of the y − axis, divide the region into elementary vertical strips (thin rectangles) of height y and width D x . Then, A is the limit sum of the areas of the vertical strips. Hence, we get A = lim −∆