7.
Application of Definite Integrals
I
1) Area under the Curve :
Y
al
·
-
y = f(x) x =
F(y]
Y =a
[ >
C
>
x = a S= b
Area bounded by the curve y = F(x)
,
Area bounded by the curve
x-acis and the line 1 = a
,
1) = b 1) = fly) y-axis and the
,
is y = b is given by
given by A I line y = a
,
A =
2) Standard Circle : 3) Standard Parabola
Y
!
Case 1: y
=
= ya)
y = b
Y
0 x= a X
IIIIII
. e
Equation : (2 + y
=
= 52 Case II : y
=
= -
yax
Centre = 10
,
01
Y
Radius = S
·I
x
Area under circle = 4 Area of O
SHADED Region
Case III : x2 =
nay Area under Parabola : 2 Area under
shaded region
a
4) Standard Ellipse :
[
1 Std form :
+
O
Case Iv : (
2
= -
yay If a > b If a < b
a Y Y
b
A b
[
O a) x -a)x
O
E
7 ~
I"Areaunderregion
~
Area under Ellipse : "
5) To find Area bet ween 2 curves or a curve and a line :
Steps : 1 Solve their Equations ↑ use -
substitution to find their
Point of intersection , -cordinate enough t
2) Find A Area bounded by the first curve and X-axis
A
I
31 Find A 2
I Area bounded by the second curve and X-axis
Az =
41 Required Area =
A, -
A2
↑
Q.1) find the area of the region bounded by the curve y = 6x + u
,
the x - axis and the lines x = 0 and x1 = 2 .
> Area bounded by curve 15th x-axis
y
-
:
,
and the lines x is
= 0
,
1 = 2
given by :
- (ox + 11 di
/
A
fyde
=
S
-
A (161312 11312 %
+
=
-
/(4144314 *
x3/x]
↑
= -
(2)
154 87
=+
-
A = 56 9& :
Units
-
Q.2) Find the area of the region bounded by the line 2y + x = 8
,
the X- axis and the lines x = 2 and1 = 1 .
·
I
-
2y +x = 8
8 x
2y = -
y =
(
Area bounded by curve y = k-axis and
,
2
1 = 2 7= 4
,
=9
b
·
=
=
= (8x 1
-
+ (x sy (10 2)]
: - - -
+ /24 14)
1x10
= - =
A = 5 So units
-
Q.3) Find the area of the region bounded by the curve y = (2 + 1 ,
the X-axis and the lines (1 0 and x 3
= =
> Area bounded by the curve
y <12 +1 -all is
-
=
,
and the lines (2 = 0 11 = 3
,
b
A =
fydx
A
A =
(( +
x]
=
((x2 + 1)dx .
=
(2 3) + -
0
9 + 3
=
A = 12 So-units
4) Find the area of the region bounded by the curve y = xY
X-axis and the lines x = 1
,
(2 = 5.
> Area bounded the curve y 14 ce-allis
by
-
=
, ,
and the lines = 1 x = s
,
b
A =
fydx
a
A =
(x]
ju4m
5
= 1
-
=
35-1
soiti
3
= =
5 Find the area of region bounded by curve y = 2 -
x2 , X-axis
R And the lines x =
-1 x = 1
,
let F(() = 2 -
12 Area bounded
by Curve y = 2-112 ,
1) assis and line x = 0 x1 =
1
,
F( ()
-
= 2 -
( -
1) I
2 f2
F( x]
- = 2 -
x2 A = - xdx
#( x) -
= f(x)
It is an even function
O
=
2-%
I f(x))di)
I = 2
2/2 - 1 -
= :
A =
10/3 So rnil
.
i S
2Areat
Y
M
7 y2
= u7 A =
B region
(0ABO)
-
a
L
10A 7 3
Area
y = 12 I
lines
rounded
x =
~
0 ,
by were
x-axis
x = 3
and
yuu
the
,
= (2) de
s
1)(3R
A =
8 x 1313
=
3
=
A = 8 13 SQ units
.
7
a
x2 + yz = 25
E
10 = 5
Yxy2 = 25 -
12
= 25 12
y = -
T
& 12dx = + sint(
Sin" (1) sin" 10)
1 0
= =
,
unde
Area
BAmdece
a
(iMC12
-
O
be (since ,
lying
It is
Grade)
in Ast
= + sin()
A =
n+ sin()
A
125 sin() -
=
sin
(2)
7/1 1017
sin"(1)
2
:
= X -
Sin
u( * I
-
24]
scieis
XXXI
= = 25t
g
Edin of Ellipse
1
>
- :
comparing with - =
5(a(b) y2 (4 -x2)
2
a = e b = - =
a= 5
E P
22
/
y =
It x
~
eeBeinumee
s
Ama n
of ellipse
Ellipse
=u de /sinc
Ast
,
It le In
podeant]
10[I
+
1 sin()
Area of Ellipse = 10 X 1 sin" 113
2
10X "
=
= lot SQ units
.
2
& Find the area of the region bounded by the curve x = 16 y ,
LDR Y = 1 y = 4 and the y-axis lying in first Quadrant
,
> Area bounded by wave 12 16y I= 1 He
-
=
&
Y-acis and the line y = 1
,
y = 1 .
S
&
b
A =
fxby A =
ug(y)" dy = (11)
C I
115
=
/"I
=
de
xi
=
8
/(91312] ,
= =
56/3 goun
18 Find the area bet ween the t wo curves (parabola) y = = 7)
LpR and 112 = 7y
Step 1 : Coordinate of t
>
-
pr ty:
all Ly"-7x
M
12/7 in eacil
i
Substitute y =
:
74
39 8
=
7 Ex 14 - 343
~ 1 ((13 343)
-
= 8
y2
=
F1 (2 =
74
= 793 343 =0
1
11 0
Y = = -
① <3 =
343
(24 = 34371 <13 = (7)3
72 = 7
Step 1 : Find Area under 1st curve (y2 : +11)
Area bounded by curve y : te -accie and
,
line =0 1 : 7
,
A fy IE
(1312 T
=
, d Al =
A
.. E his de = 17
x&x(7)3
O
IEXXI
(ind
=
A = +F
IX2X7 I
= =
Steps : Find Area op 2nd wrve (2 ty)
=
Area bounded
12
and curve
by y =
,
-avil and
line x = 0 11 = 7
(2
,
+
D Ac :
Az =
f Yzd
a =
1x(7)3
7X3
J
=
di
19
Al =
Recevired Area =
1A, -
All
18-y
:
19 sounits
=
11 Find the area of the region bounded by the curve xy = c
I
the 1-axis and the line K = c x = 2c
,
Area bounded
e
y and
>
by curve
= , -acie
-
line < = c
,
x = 2) is given by
(10911)"
b
A
fyd) A 12
= =
A
= c (10g 22 -
1091)
= =
x x log(24
12 X
A =
log2 Scwnm -
12
: +
-
A -2xdx 2xdx
I
-
I ! I
b
&
foxedx Yfxxdx y ( xx ( +
=
=
+ + x)xx
a a
10 1)7 /(2)2-(02)
gyd +
= - 1 -
+
A
: = 1 +4
A = 5 sornits
Y
&
3 &
2 a
a
I
- 2
u
-
-
