We know the method(s) of finding areas of triangle, quadrilateral, polygon, circle etc. In this chapter we shall learn how to find the area of the regions bounded by curves of the following types.

Type I: y = f(x), the x-axis and the lines x = a and x = b where f is a continuous function in [a, b].

Type II: The curves y = f1(x), x ∈ (a, c), y = f2(x) , x ∈ (c, b), the x-axis and the lines x= a, x = b ; a < c< b . .

Type III: The curves y = f1(x), y = f2(x) and the lines x = a, x = b.

Type IV: Bounding curves are represented by function defined through given conditions.

Type V: Miscellaneous

(i) The area bounded by y = f(x) above the x- axis and the coordinates x = a and x = b (represented by shaded portion ) is given by

Calculate the area bounded by the parabola y = x2, x-axis, and the lines x =2, x=4. .

= square units.

The area of the shaded portion is given by

, where x = c is a solution of f1(x) = f2(x), a< c< b.

Find the area included between the curves y = sin-1 x , y = cos-1x and the x-axis.

Clearly we have to find the area of the shaded region OPBO. The point P of intersection of .

y = sin-1x and y = cos-1x is obtained by solving

sin-1x = cos-1x

&⇒ p/2 - cos-1x = cos-1x

&⇒ 2 cos-1x = p/2

&⇒ cos –1x = p/4

x =

The required area

D =

=

=

Put x = sinq

dx = cosq dq

x = 0, q = 0

x = 1/, q = p/4

x = 1, q = p/2

&⇒ The required area

D =

=

=

= sq. units.

The shaded area, which is bounded by two curves y = f1 (x), y = f2(x) and the ordinates

x = a, x= b is given by

.(*)

Note: If y= f1(x) is above the x-axis and

y = f2(x) is below the x-axis, or y = f1(x) and

y = f2(x) are both below the x-axis or partly above and partly below the x-axis, even then formula (*) works. .

Find the area included between the line y = x and the parabola x2 = 4y.

The line y = x cuts the parabola at O and B whose x coordinates are 0 and 4 respectively

The required area =

= =

= = sq. units.

In this type we consider those cases, when the function is not given explicitly. Functions representing the curves are to be obtained or formulated from the given condition. This is illustrated by the following example. .

Find the area bounded by f(x) = maximum{sinx, cosx}, x= 0 , x = 2p and the x-axis.

Solution:

The graph of the function y = f(x) is shown by bold curves. .

The required region is shaded.

The required area

=

= = Ö2+ 4 sq. units.

In this type we consider those cases when a variable area is bounded by different curves (One curve unknown) is given, and we have to find the unknown curve y = f(x)

If the area bounded by the x-axis, the curve y = f(x) and the lines x = a, x = b is equal to , " a < b, where ‘a’ is a given positive real number, find f(x).

According to the given condition

Differentiating with respect to b (as b is a variable), we get

f(b) = ±.2b

f(x) = ±