Integral Calculus

Fig. 4.1 represents an arbitrary function y of x, and the area of the region bounded by the curve, the x-axis and the verticals x = x₀ and x = x₁ is designated A. What is the increase δA in this area if x₁ is increased to x₁ + δx?Apart from the small approximately triangular area that is shaded in the diagram, all of the increase in area is accounted for by the tall thin rectangle of height y₁ and width δx,which has area y₁δx. In symbols, If x is made smaller, the small shaded triangle decreases rapidly in area, not only absolutely, but also in relation to the area of the rectangle. Consequently, the approximation embodied by the above equation is very accurate if δx is very small, and in the limit as δx approaches zero it becomes exact.