(i) Let y = f(x) and y = g(x) are two curves, which intersect each other at .t = a and x = b as shown in figure. For finding the area between the two curves, we take a vertically thin strip whose length is f(x) - g(x)] and breadth is dx, then area of this vertical strip is :
dA = [f(x) - g(x)] dx
And, the total area between the two curves is taken as:
A = \(\int^b_a\) [f(x)-g(x)]dx
We can also find the area between two curves as :
A = [area bounded by y = f(x), j-axis and the lines x = a, x = b] - [area bounded by y = g(x), x-axis and the lines x - a, x = b]
A = \(\int^b_a\) f(x) dx - \(\int^b_a\) g(x) dx = \(\int^b_a\) a[f(x)-g(x)] dx where f(x) ≥ g(x) in [a, b].
(ii) In interval [a, b],f(x) > g(x)
If c be any point between a and b(a< c<b), where both the curves intersect each other as shown in figure, then the area of the regions bounded by curves can be obtained as :
Required area = Area of region APBQA + Area of region BSCRB
= \(\int^c_a\) [f(x)-g(x)] dx + \(\int^b_c\) [g(x) - f(x)] dx
∴ f(x) ≥ g(x) in [a, c] and f(x) < g(x) in [c, b] where a< c < b.
→ The area of the region bounded by the curve y = f(x), x-axis and the lines x = a and x = b is given by the formula:
Area = \(\int^b_a\) y dx = \(\int^b_a\) f(x) dx
→ The area of the region bounded by the curve x = y-axis and the lines y = c, y = d is given by the formula:
Area = \(\int^d_c\) g(x) \(\int^b_c\) (y) dy
→ The area of the region enclosed between two curves y = f(x), y g(x) and the lines x = a, x = b is given by the formula:
Area \(\int^b_a\) |f(x) - g(x)|dx where f(x) ≥ g(x) in [a, b].
