(4) Let f and g be two continuous real valued functions on [a, b] and c ∈ (a, b) such that
f(x) < g(x), ∀x ∈ [a, c) and g(x) < f(x), ∀x ∈ (c, b] with
f(c) = g(c).
Then the area (A) of the region bounded by y = f(x), y = g(x) and
the two lines x = a, x = b is given by
A |
= |
[g(x) – f(x)] dx + [f(x) – g(x)] dx |
|
= |
|(f(x) – g(x)) dx| + |(f(x) – g(x)) dx| |
Refer adjacent fig. (iv).
(5) If a region is bounded by the curves with equations
x = f(y), x = g(y), y = c and y = d,
where f and g are continuous and f(y) ≥ g(y) for c ≤ y ≤ d,
then its area is given by
A = [f(y) – g(y)] dx
In general, when f(x) ≥ g(x) on [c, d]
or f(y) ≤ g(y) on [c, d], we have
A = |(f(y) – g(y)) dx|
Refer adjacent fig. (v).