APPLICATION OF INTEGRALS A.L. Cauchy ( - ) APPLICATION OF INTEGRALS Fig . This area is called the elementary area which is located at an arbitrary position within the region which is specified by some value of x between a and b. We can think of the total area A of the region between x -axis, ordinates x = a , x = b and the curve y = f ( x ) as the result of adding up the elementary areas of thin strips across the region PQRSP.
Symbolically, we express A = A ydx f x dx The area A of the region bounded by the curve x = g ( y ), y -axis and the lines y = c , y = d is given by A = xdy g y dy Here, we consider horizontal strips as shown in the Fig . Remark If the position of the curve under consideration is below the x -axis, then since f ( x ) < from x = a to x = b , as shown in Fig . , the area bounded by the curve, x -axis and the ordinates x = a, x = b come out to be negative. But, it is only the numerical value of the area which is taken into consideration.
Thus, if the area is negative, we take its absolute value, i.e., a f x dx Fig . Generally, it may happen that some portion of the curve is above x -axis and some is below the x -axis as shown in the Fig . . Here, A < and A > .
Therefore, the area A bounded by the curve y = f ( x ), x -axis and the ordinates x = a and x = b is given by A = |A | + A . Example Find the area enclosed by the circle x + y = a . Solution From Fig . , the whole area enclosed by the given circle = (area of the region AOBA