Application of definite integral for determination of areas (simple cases).

Let y = f_{1 }(x) and y = f_{2 }(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 = f_{1}(x) and y = f_{2}(x) between
the ordinates x = a and x = b (i.e. the area of the region A_{1}B_{1}C_{1}D_{1}). .

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

Clearly AD @ BC = f_{1}(x) ~ f_{2}(x).

Now the area of this strip,

dA = (f_{1}(x)~f_{2}(x))
dx

Thus the required area,

A=

Remarks :

- If f
_{1}(x) ≥ f_{2}(x) " x ∈ [ a, b] then A = - If f
_{1}(x) £ f_{2}(x) " x ∈ [ a, b] then A = - If f
_{1}(x) ≥ f_{2}(x) " x ∈ [ a, c] - and f
_{1}(x) £ f_{2}(x) " x ∈ [ c, b], - then =
- We can finally
conclude that the area bounded by the two curves between

x = a and x = b is - If f
_{2}(x) = 0 " x ∈ R, - would give us area
bounded by y = f
_{1}(x) and the x-axis between the ordinates x = a and x = b (here we are assuming that f_{1}(x) ≥ 0 " x ∈ [ a, b] ).

Similarly would give us the area bounded by the
curves x = f_{1}(y) and x = f_{2}(y) between the lines y =
c and y = d