AREAS OF BOUNDED REGIONS
109
APPLICATIONS OF INTEGRALS 6.1
AREA UNDER SIMPLE CURVE 1. The area of the region bounded by a curve y = f(x), x =a, x = b, and the x - axis. Case – I When the curve y = f(x) lies above the x-axis. The area bounded by the curve y = f(x), the x-axis b
y dx
and the ordinates x = a and x = b is given by
a
Y y = f(x)
B
A
X’
C
D b
a
O
X
Y’
Case – II
When the curve y = f(x) lies below the x-axis.
The area bounded by the curve y = f(x), the x-axis, b
and the ordinaes x = a and x = b is given by
(y) dx a
Y X’
C
O
b B
a A
X
D y=f(x) Y’
2.
C
The Area of the Region Bounded by a Curve x = f(y), The Abscissae y = c, y = d and the y -axis. Case – I When the curve x = f(y) lies to the right of the y -axis. The area bounded by the curve x = f(y), the y-axis d
and the abscissae y = c and y = d is given by
x dy c
Y y=d B
D x=f(y) C
A
y=c
X’
X Y’
Case – II When the curve x = f(y) lies to the left of the y-axis. The area bounded by the curve x = f(y), the y-axis d
and the absissa y = c and y = d is given by
(x) dy c
www.thinkiit.in
AREAS OF BOUNDED REGIONS
111 Y B(0,a) a a
X’
X A(a,0)
O (0,0)
Y’
(iii)
centre at (, ) and radius = a. x2 + y2 + 2gx + 2fy + c = 0 represents the general equation of a circle with centre at (– g, – f ) and radius =
2. (i)
Y
g2 f 2 c
Parabola y2 = 4ax, where a > 0 or y2 = 4ax, whre a < 0 (standard equation) X’ It is symmetric about x -axis, where O is the vertex, F is the focus, LL is the latus rectum :
Y
L
x=a Latus rectum
F
X
O
F
X’
x=–a
L’
L’
Y’
Y’
y2 = 4ax, a > 0
LL' X' X; X' X being the axis of the parabola. (ii)
X
O
Latus rectum
x2 = 4by, where b > 0 or x2 = 4by, where b < 0
y Latus rectum
It is symmetric about y-axis, where LL is the latus rectum.
y=b
F L
L’
Y
X’
O
X’
X
O
X
Y’ x2 L’
= 4by, b > 0
L y=–b
F
Latus rectum Y’ x2 = 4by, b < 0
O is the vertex, F is the focus, LL is the latus rectum, LL' YY' . Ellipse
3.
x2 y2 1 represents an ellipse with centre at (0, 0). a2 b2 It
is
symmetric
about
both
the
axes,
meeting
Y B(0,b) A’(–a,0) X’
A(a,0) O (0,0)
X
B’(0,–b) Y’
and y-axis at (0, ±b) and a > b. Here, AA = 2a = length of major axis. BB = 2b = length of minor axis.
www.thinkiit.in
x-axis
at
(±a,
0)
AREAS OF BOUNDED REGIONS
113 4
x 9 16 4
x2 Y
x2=4y
A(– 2 ,1/ 2)
B( 2 ,1/ 2) 4x2 + 4y2 = 9
X’
X
O
– 2
2
Y’
x4 + 16x2 – 36 = 0 (x2 + 18) (x2 – 2) = 0 x2 = 2 or – 18
x=± 2 (x2 cannot be – 18) y
1 x2 = 2 4
Taking y =
9 x2 4
Required area =
on f(x) and y =
2
– 2
x2 as g(x), where f(x) g (x) in [– 2 , 2 ] we have ve 4
2
[ f ( x ) g( x )] dx = 2 [ f ( x ) g( x )] dx = 2 0
2 0
9 x2 x 2 dx 4 4
2
9 9 2 x2 2 3 x 9 2 x x 9 2 2x 4 4 sin 1 – sin 1 – = 2 = 2 2 8 2 8 6 8 12 8 0 0
2
2 2 2 9 2 2 2 2 9 2 2 2 9 = sin1 = 2 sin 1 = 2 sin 1 3 sq. units 3 6 4 3 12 8 8 8 4 Ex.5 Sol.
Find the area bounded by curves (x – 1)2 + y2 = 1 and x2 + y2 = 1. Given equations to the curves are x2 + y 2 = 1 ...(1) Y (x – 1)2 + y2 = 1 ...(2) (x – 1)2 + (1 – x2) = 1
x2 – 2x + 1 + 1 – x2 = 1 2x = 1 x =
Requried area 1 1/ 2 2 y of circle (2) dx y of circle (1) dx 0 1/ 2
1 2
X’
1 1/ 2 2 1 ( x 1)2 dx 1 x 2 dx 0 1/ 2
O
(1,0)
Y’ x=1/2 x=1
1/ 2 1 2 x 1 x2 1 1 1 ( x 1) 1 ( x 1) 1 2 sin ( x 1) sin x ) 2 2 2 2 1/ 2 0
1 1 1 1 1 1 1 1 1 1 2 1 sin 1 0 sin 1( 1) 0 sin 1(1) 1 sin 1 4 2 2 2 4 4 2 2 2 4
www.thinkiit.in
X
AREAS OF BOUNDED REGIONS
115
UNSOLVED PROBLEMS EXERCISE – I Q.1
Using integration, find the area of the region bounded by the lines y = 4x + 5, y = 5 – x and 4y – x = 5.
Q.2
Find the area of the region bounded by the curve x2 = 4y and the lines x = 4y – 2
Q.3
Find the area of the region bounded by the curve y = x2 + 2, y = x, x = 0 and x = 3
Q.4
Find the area of the region enclosed between the parabola y2 = x and the line x + y = 2 in the first quadrant.
Q.5
Find the area of the smaller part of the circle x2 + y2 = a2 cut off by the line x =
Q.6
Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2 = 32.
Q.7
Find the area of the region bounded by the curve y 1 x 2 , the line y = x and the positive
a 2
x-axis. x y x2 y2 2 1 and the line 1 2 a b a b
Q.8
Find the area of the smaller region bounded by the ellipse
Q.9
Find the area included between the curves y2 = 4ax and x2 = 4ay, a > 0
Q.10
Draw the rough sketch of y2 = x + 1 and y2 = – x + 1 and determine the area enclosed by the two curves.
Q.11
Calculate the area of the region enclosed between the circle x2 + y2 = 4 and (x – 2)2 + y2 = 4
Q.12
Using integration, find the area of the circle x2 + y2 = 16 which is exterior to the parabola y2 = 6x
Q.13
Sketch the region common to the circle x2 + y2 = 16 and the parabola x2 = 6y. Also find the area of the region, using integration 0
Q.14
Sketch the graph of y = |x + 3|. Evaluate
| x 3 | dx What does the value of the integral
–6
represent on the graphs ? Q.15
Using integration find the area of the region bounded by the following curves after making a rough sketch : y = 1 + |x – 1|, x = – 3, x = 3 and y = 0
Q.16
Find the area of the region {(x, y) : x2 + y2 1 x + y}
Q.17
Find the area of the region {(x, y) : x2 y |x|}
Q.18
Find the area of the region : {(x, y) : 0 y (x2 + 1), 0 y (x + 1)}, 0 x 2}
Q.19
Find the area of the region : {(x,y) : x2 + y2 2ax}, y2 ax, x 0, y 0
Q.20
Find the area bounded by the curve y = sin x between x = o and x = 2.
ANSWER KEY 1.
9.
15 2
16a2 3
15. 16
2.
10.
9 8
3.
21 4. 2
8 3
1 16. 4 2
7 a 2 1 5. 6 2 2
8 2 3 11. 3
17.
1 3
18.
6. 4
7.
12.
23 6
8
4 3 8 3
8.
13.
16 4 3 3
a 2 2a 2 19. 4 3
www.thinkiit.in
ab ab – 4 2 14. 9
20. 4
117
AREAS OF BOUNDED REGIONS
www.thinkiit.in
