AREAS
AREAS

AREAS

AREAS
 In the previous chapter we have learnt the concept of
definite integration which can be used to find the
areas of the region bounded by various parameters.
 In this chapter we study the area bounded by the
graph of a non-negative continuous function y=f(x)
defined on [a,b] and the x-axis as an application of the
definite integral.

AREAS
 Depending upon the nature of the curves, the area can
have different shapes and thus the tool of definite
integral can be employed to find the areas of different
shapes.
 The application of areas is done in the various fields like
health, management, photography, geology etc.

AREAS
1. The area of the region bounded
by the curve
AREA OF BOUNDED REGIONS
x=b
O
x
x=a
y
y=f(x), x-axis
and the ordinates x=a, x=b
is a
b
f(x) dx if f(x)  0  x  [a,b]

AREAS
x
x′
O
y
y′
y=f(x)
2. The area of the region bounded
by the curve
x=a x=b
y=f(x), x-axis
and the ordinates x=a, x=b
is - a
b
f(x) dx
(or)
a
b
f(x) dx if f(x) ≤ 0  x  [a,b]

AREAS
3. The area of the region
bounded by the curve
y=a
y=b
x
O
y
x=f(y)
and the lines y=a, y=b and y-axis is
𝒂
𝒃
f y dy if f(y) ≥ 0,∀y ∈ [a,b]
x=f(y)

AREAS
4) The area of the region bounded by
the curve x=f(y)
y=a
y=b
x O
y
x=f(y)
and the lines y=a,
y=b and y-axis is
𝒂
𝒃
−𝒇(𝒚)𝒅𝒚
(or)
𝒂
𝒃
𝒇 𝒚 𝒅𝒚 , if f(y) ≤ 0 ∀ y ∈ [a,b]

AREAS
5. Let y=f(x) be continuous on
[a,b] such that f(x)≥0 x[a,c]
Then the area of the region
bounded by the curve y=f(x)
the x-axis and the lines x=a
x
a
xb
y  f(x)
y
O
x
a
b
c
and f(x)0 x[c,b] where a<c<b.
and x=b is given by
A = 𝒂
𝒄
f(x) dx + 𝒄
𝒃
f(x) dx

AREAS
and the lines y = a and
y = b
6. Let x=f(y) be continuous on [a,b]
such that
y=a
y=b
y
O
x
c
x=f(y)
A 
a
c
f(y) dy +
c
b
f(y) dy
f(y) ≥0  y  [a,c] and
f(y)  0  y  [c,b] where a<c<b.
Then the area of the region
bounded by the curve x=f(y),
the y-axis
is given by

AREAS
then the area of the region
enclosed by the curves between their points of
intersection is given by
AREA OF THE REGION BOUNDED
BY THE TWO CURVES
y=f(x)
y
O
x
y=g(x)
x1 x2
a b
7. Let y=f(x) and y=g(x) be two
continuous functions on a, b
such that they intersect at x1,
x2(a,b) and x1  x2.
If f(x)  g(x)  x  (x1,x2)
upper curve – lower
curve
[f(x)-g(x)] dx
A = 
x1
x2

AREAS
be two
continuous functions in [a,b]
8. Let y=f(x)
a
c
f(x) − g(x dx
y=f(x)
y
O
x
y=g(x)
a b
c
+
c
b
g(x) − f(x dx
and intersect at x=c[a,b]
then the area between the curves is
and y=g(x)

AREAS
9. Let the curves y  f(x)
Area between the given curves
(Non- Intersecting curves)
y=g(x)
O
x
y=f(x)
x  b
y
x  a
and y  g(x)
be such that f(x) ≥ g(x) ≥ 0  x  [a,b]
then the area bounded by the curves
and the lines x=a, x=b is
upper curve – lower
curve
[f(x)-g(x)] dx
A = 
a
b

AREAS
1) The area of the region bounded by the
curve y=f(x), x-axis and the ordinates x=a,
x=b (a>b) is
1) a
b
f(x) dx
2) a
b
f(y) dy
3) b
a
f(x) dx
4) b
a
f(y) dy

AREAS
2) The area of the region bounded by the curve x=f(y) and
the lines y=a, y=b and y-axis (a>b) is
1) 𝒂
𝒃
f x dx
2) 𝒃
𝒂
f x dx
3) 𝒂
𝒃
f y dy
4) 𝒃
𝒂
f y dy

AREAS
3) Let y = f(x) and y = g(x) be two continuous functions in
[a,b] and intersect at x=c[a,b], f(x) ≥ g(x) ∀ x ∈ a,c , f(x) ≤
g(x) ∀ x ∈ [c,b] then the area between the curves is
1) c
a
f(x) − g(x dx + c
b
g(x) − f(x dx
2) a
c
f(x) − g(x dx + c
b
g(x) − f(x dx
3) a
c
f(x) − g(x dx + b
c
g(x) − f(x dx
4) c
a
f(x) − g(x dx + b
c
g(x) − f(x dx

AREAS
4) Let the curves y=f(x) and y=g(x) be such that f(x)  g(x)  0
 x  [a,b] then the area bounded by the curves and the
lines x=a, x=b is
1)
𝒃
𝒂
[f(x)−g(x)] dx 2)
𝒃
𝒂
[f(y)−g(y)] dy
3)
𝒂
𝒃
[f(y)−g(y)] dy 4)
𝒂
𝒃
[f(x)−g(x)] dx

AREAS
5) The area under the curve y=sinx in [0,2] is ___ sq. unit
1) 2
2) 4
3) 8
4) 16

AREAS
6) The area in sq. units bounded by the x-axis, part of the
curve y=1+
8
x𝟐 and the ordinates x=2 and x=4 is
1) 4
2) 2
3) 12
4) 16

AREAS
7) The area bounded by y=x2+2, x-axis, x=1 and x=2 is
1)
16
3
2)
17
3
3)
13
3
4)
20
3

AREAS
8) The area bounded by the parabola x=4-y2 and the y-axis
in square units is
1)
3
32
2)
32
3
3)
33
2
4)
16
3

AREAS
Thank you…