The document discusses different types of bounded regions and calculating their areas using integrals. It defines three types of regions: 1) bounded by two curves and vertical lines, 2) bounded by two curves, and 3) bounded by modulus functions where one curve is greater than the other over some intervals. Examples are provided for each type, such as finding the area between a parabola and line, two parabolas, a parabola and circle, and two circles. The key idea is that the region's area can be expressed as a definite integral of the differences between the bounding curves.

1. - AREAS OF BOUNDED REGIONS
Himani Asija
Delhi Public School
Vasant Kunj

2. THEOREM : Let f(x) be a
•continuous function defined in [a,b].
Then the area bounded by
the curve ,
the x axis and
the ordinates a and b
is given by
b b
∫ f ( x)dx = ∫ ydx
a a
or
b b
∫ f ( y )dy = ∫ xdy
a a

3. With different values of n, let’s see what the area looks like !
def int 3

4. Types of integrals to be evaluated :
y
y=f(x) Type 1 region :
S: Bounded by two curves y=f(x) and
g(x) and between two vertical line y=a
and y=b.
S
o a b x S = {( x, y ) | a ≤ x ≤ b, g ( x) ≤ y ≤ f ( x)}
b
y=g(x)
A = ∫ [ f ( x) + {− g ( x)}]dx
a
OR
y
y=f(x) S: Bounded by two curves y=f(x) and
g(x) and between two vertical line y=a
and y=b.
S
S = {( x, y ) | a ≤ x ≤ b, g ( x) ≤ y ≤ f ( x)}
y=g(x)
b
o a b x A = ∫ [ f ( x ) − g ( x )]dx
a

5. Type 2 region : S: Lies between two curves x=f(y) and
g(y) and between two line x=c and y=d.
y
y=d
d S = {( x, y ) | c ≤ y ≤ d , g ( y ) ≤ x ≤ f ( y )}
d
x=g(y) x=f(y)
A = ∫ [ f ( y ) − g ( y )]dy
c
S
c
y=c
x
o

6. Type 3 region : modulus functions or functions
when f>g in one part of interval and g>f in the other
part
To find the area between the curves y=f(x) and y=g(x)
where f(x)>g(x) for some values of x but g(x)>f(x) for
other values of x, then we split the given region into
several regions S1, S2, …with areas A1, A2, ….,

7. Type 3 region : modulus functions or functions
when f>g in one part of interval and g>f in the other
part
To find the area between the curves y=f(x) and y=g(x)
where f(x)>g(x) for some values of x but g(x)>f(x) for
other values of x, then we split the given region into
several regions S1, S2, …with areas A1, A2, ….,
s2 s3
π /4 π /2
s6
s4
s5

8. TYPES OF QUESTIONS
TYPE 1 AREA BOUNDED BETWEEN A CURVE AND A LINE
EXAMPLE: Find the area bounded by the parabola x²=4y and the line
x=4y-2
def int

9. EXAMPLE 2 Find the area of the region {(x,y): x²≤y ≤x}
def int

10. TYPE 2 AREA BOUNDED BETWEEN TWO CURVES
(a) Between 2 parabolas
EXAMPLE: Find the area bounded by the parabola y²=4ax and x²=4ay,
a>0
def int

11. (b) Between a parabola and a circle
EXAMPLE 1 : Find the area of the region {(x,y) : y² ≤ 4x, 4x²+4y² ≤ 9}

12. EXAMPLE 2 : Find the area of the region {(x,y) : y² ≥ 4x, 4x²+4y² ≤ 9}
def int

13. (c) Between 2 circles
EXAMPLE: Find the area bounded by the circles x²+y²=1 and (x−1)²+y²=1

14. (c) Between lines
EXAMPLE: Using integration, find the area of triangle whose
vertices are A(2,5) B(4,7) C(6,2)

15. def int

16. 5π / 4
3π / 2
π/ 2
π /4