# Area Bounded Region

### BASIC CONCEPTS :

Let y = f1 (x) and y = f2 (x) be two given curves, which are continuous in [a, b]. Suppose we have to find the area of the plain region bounded by the two curves y = f1(x) and y = f2(x) between the ordinates x = a and x = b (i.e. the area of the region A1B1C1D1).

For this purpose we take an elemental strip ABCD of width dx and height AD (or BC).

Clearly AD ≅ BC = f1(x) ~ f2(x).

Now the area of this strip,

dA = (f1(x)~f2(x)) dx

Thus the required area,

$\displaystyle A = \int_{a}^{b} (f_1(x) \sim f_2(x)) dx$

Remarks:

If f1(x) ≥ f2(x) ∀ x ∈ [ a, b] then

$\displaystyle A = \int_{a}^{b} (f_1(x) - f_2(x)) dx$

If f1(x) ≤ f2(x) ∀ x ∈ [ a, b] then

$\displaystyle A = \int_{a}^{b} (f_2(x) - f_1(x)) dx$

If f1(x) ≥ f2(x) ∀ x ∈ [ a , c] and f1(x) ≤ f2(x) ∀ x ∈ [ c , b] , then

$\displaystyle = \int_{a}^{c} (f_1(x) - f_2(x)) dx$ + $\displaystyle \int_{c}^{b} (f_2(x) - f_1(x)) dx$

We can finally conclude that the area bounded by the two curves between x = a and x = b is

$\displaystyle \int_{a}^{b} (curve \; lying \; above - curve \; lying \; below) dx$

If f2(x) = 0 ∀ x ∈ R,

$\displaystyle \int_{a}^{b}f_1(x) dx$

would give us area bounded by y = f1(x) and the x–axis between the ordinates x = a and x = b (here we are assuming that f1(x) ≥ 0 ∀ x ∈ [ a, b] ).

Similarly

$\displaystyle A = \int_{a}^{b} (f_1(y) \sim f_2(y)) dy$

would give us the area bounded by the curves x = f1(y) and x = f2(y)
between the lines y = c and y = d.

### TYPES OF PROBLEMS :

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

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