In the figure shown below, y = f(x) is a curve. For finding the area bounded by the curve y = f(x), x-axis and
the ordinates x = a and x = b, we think of the area under the curve as composed of lines number of very thin vetical strips. Consider an arbitrary strip PQRS of height y and width dx, then area of this strip dA = ydx. 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.
Total area A of the region between x-axis, ordinates x = a, x = b and the curve y =/(x), will be the result of adding up elementary areas of thin strips. So,
A = \(\int^b_a \) dA = \(\int^b_a \) y dx = \(\int^b_a \) f(x) dx
Similarly,-for finding the area bounded by the curve x = g(y), y-axis and lines y = c and y = d, we divide the area under the curve as composed as large number of very thin horizontal strips as shown in figure below.
If this area is denoted by A, then
A = \(\int^d_c\) xdy = \(\int^d_c\) g(y) dy
Here, we consider horizontal strip.
Remark:
If the position of the curve under consideration is below the x-axis, then since f(x) < 0 from x = a to x = b as shown below, 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., |\(\int^b_a\) f(x)dx|
Generally, it may happen that some portion of the curve is above x-axis and some is below the x-axis as shown below. Here, A1 < 0 and A2 >0. Therefore, the area
A bounded by the curve y = f(x), x-axis and the ordinates xa and xb is given by A
The Area of the Region Bounded by a Curve and a Line:
With the above learnt method, we can find the area of the region bounded by a line and a circle, a line and a parabola, a line and an ellipse. Equations of above mentioned curves will be in their standard forms only as the cases in the other forms go beyond the scope of this textbook.
