LPP - AREA

SUBJECTIVE
1. Find the area of the region {(x, y) : 0  y  x2 + 1, 0  y  x + 1, 0  x  2}.

2. Find the area of the region formed by x2+ y2 – 6x – 4y + 12  0, y  x and x  5/2.

3. A chord is drawn to the curve x2 + 2x – y + 2 = 0 at the point whose abscissa is 1, and it is parallel to the line y
= x. Find the area in the first quadrant bounded by the curve, this chord and the y-axis.

4. Find the area between the curves y = x2 + x –2 and y = 2x, for which
|x2 + x –2| + | 2x | = |x 2 + 3x –2| is satisfied.

5. (i) Show that the area included between the parabolas y2 = 4a(x + a) and y2 = 4b(b – x) is 8 / 3 ab (a+b) .
(ii) Find all the possible values of b > 0, so that the area of the bounded region enclosed between the
parabolas y = x – bx2 and y = x2/b is maximum.
b f b 

 f  x  dx    f x  dx  bf b   af  a  .
1 
6. If a function f (x) is bijective in [a, b], then prove that
a f a

7. If the area enclosed by the parabolas y = a – x2 and y = x2 is 18 2 sq. units. Find the value of 'a'.

8. The area enclosed by the curve f (x) = 24 + ax  x2, co-ordinate axes and the ordinate x = 6 is 108. If m and n
are x-axis intercepts of the graph of y = f (x), then find the value of m + n + a.

9. Let f(x) be a polynomial of degree three such that the curve y = f(x) has relative extremes at
2
x= and passes through (0, 0) and (1, –1) dividing the circle x2 + y2 = 4 in two parts. Find the ratio of the
3
areas of two parts.

10. Let c1, c2 and c3 be the graphs of the function y = x2, y c2 c1

y = 2x, y = f(x), 0  x  1, f(0) = 0. For a point P on c1, let
the lines through P, parallel to the axes meet c2 and c3 at
Q
Q and R respectively (see figure). If for every position P c3
P(on c1) the area of the shaded region OPQ and ORP are
equal, determine the function f(x). O x
R

11 Consider the collection of all curve of the form y = a – bx2 that pass through the point (2, 1), where a and b are
positive constants. Determine the value of a and b that will minimise the area of the region bounded by y = a –
bx2 and x-axis. Also find the minimum area.

12. O(0, 0), A(2a, 0) and B a,  

3 a are three points and P(, ) is a point such that d(P, OA)  minimum [d(P,
OB), d(P, AB)] and (0    2a). Where d(P, OA) represents the distance of point P from the line OA. Find the
area of the region described by P.

13. Let An be the area bounded by the curve y = (tan x)n & the lines x = 0, y = 0 & x = /4. Prove that for n > 2, An +
An–2 = 1/(n – 1) & deduce that 1/(2n + 2) < An < 1/(2n – 2).
14. Find the area of the region bounded by y = (x – 4)2, y = 16 – x2 and the x– axis.

 e (logsec t  sec
t 2
15. Given f(x) = t)dt ; g(x) = –2ex tan x. Find the area bounded by the curves
0

y = f(x) and y = g(x) between the ordinates x = 0 and x = .
3

Corporate Branch Office : Noida Sec-62 Branch : C-56/33, Institutional Area, Sector-62, Noida
 2
16. (a) Find the region bounded by the curves y = x2 and y 
1  x2
(b) Find the area bounded by the curves 4y = |x2  4| and y + |x| = 7.

 1
17. Let f(x)  max sin x,cos x,  then determine the area of region bounded by curves y=f(x) x-axis, y-axis and
 2
x= 2 .

18. Find the area of the region bounded by the curves y=x 2 and y = sec1[sin2x], where [.] denotes G.I.F.

19. If the area bounded by the curve y=x2+1 and the tangents to it drawn from the origin is A, then the value of 3A
is

20. If the area enclosed by the curve y  x and x=  y , the circle x2+y2=2 above the x-axis, is A then the value
16
of A is


MULTI CHOICE SINGLE CORRECT

1. The area of the smaller region bounded by the circle x2 + y2 = 1 and |y| = x + 1 is
 1   
(A)  (B)  1 (C) (D)  1
4 2 2 2 2
2. The area bounded by the curve y = x2 + 2x + 1, the tangent at (1, 4) and the y-axis is
(A) 1 (B) 1/2 (C) 1/3 (D) 1/4

3. The area of the smaller portion of the circle x2 + y2 = a2 cut off by the line x=a/2 (a > 0) is
 a2  3
(A) sq. units (B) a2    sq. units
3  3 2 
 3  
(C) a2    sq. units (D) a2   3  sq. units
 3 4  2 

x
4. The area bounded by y = , x  0 the y-axis and the curve y = x3 is
x
(A) 3 3 (B) 3/2 (C) 2/3 (D) 2 3

5. The ratio in which the area bounded by the curves y2 = x and x2 = y is divided by the line x = 1/2 is
(A) (4 2 – 1) : (9 – 4 2 ) (B) (3 2 +3) : (9 – 4 2 )
(C) ( 2 – 1) : ( 3 – 1) (D) (2 2 – 1) : (3 3 – 1)

6. The area common to the circle x2 + y2 = 64 and the parabola y2 = 12x is equal to
(A)
16
3

4  3  (B)
16
3

8  3  (C)
16
3
4  3  (D) none of these

7. The area of bounded by the curve y = log x, the x-axis and the line x = e is
 1
(A) e sq. units (B)  1   sq. units (C) (2e  1) sq. units (D) 1 sq. unit
 e

8. The area of the region bounded by the curve x = sin–1 y, the x-axis and the lines |x| = 1 is
(A) 2 – 2 cos 1 (B) 1 – cos 1 (C) 1 – 2 cos 1 (D) none of these

 
9. Let f (x) = min [tan x, cot x, 1/3]  x   0,  . Then the area bounded by y = f (x) and the x-axis is
 2
4   2   4 
(A) ln    (B) ln   (C) ln    (D) none of these
3 6 3  3  12 3  3  12 3

10. The area of the region bounded by the curves y = |x| and y = 3 – |x| is
(A) 9/4 (B) 3 (C) 9/2 (D) 9

Corporate Branch Office : Noida Sec-62 Branch : C-56/33, Institutional Area, Sector-62, Noida
11. The area of the circle (x – 2)2 + (y – 3)2 = 32 below the line y = x + 1 is
32
(A) (B) 32 (C) 16 (D) none of these
3

12. If Am represents the area bounded by the curve y = ln x m, the x-axis and the lines x = 1 and x= e, then Am+ m
Am-1 is
(A) m (B) m2 (C) m2/2 (D) m2 – 1

13. The area bounded by the x-axis, the curve y = f(x) and the lines x = 1, x = b is equal to b2  1  2 for all b >
1, then f(x) is
x
(A) x  1 (B) x  1 (C) x2  1 (D)
1  x2

14. The area {(x, y) : x2  y  x } is equal to

1 2
(A) (B) (C) 1/6 (D) none of these
3 3

15. The area bounded by the curve |x| = cos–1y and the line |x| = 1 and the x-axis is
(A) cos 1 (B) sin 1 (C) 2 cos 1 (D) 2 sin 1

 1
loge | x |, | x |
e
16. Area bounded by the curve f (x) =  and x axis is
1
| x | 1  , 1
| x |
 e e
1 2 1 2 1 2
(A) 2  2  (B) 2  2  (C) 2  (D) none of these
e e e e e e

17. Area common to the curves y2 = ax and x2 + y2 = 4ax is equal to

a2 a2
(A) (9 3 + 4) (B) (9 3 + 4)a2 (C) (9 3 – 4) (D) none of these
3 3

18. Let f(x) = x2, g(x) = cos x and h(x) = f(g(x)). Area bounded by y = h(x) and x-axis between x=x1 and x=x2,
where x1 and x2 are roots of the equation 18x29x + 2 = 0, is equal to
  
(A) sq.units (B) sq.units (C) sq.units (D) none of these
12 6 3

19. The area bounded by the curve y = e|x|, y = e–|x|, x  0 and x  5 is

(A) e5 + e–5 + 2 sq. unit (B) e5 + e–5 – 2 sq. unit
(C) e – e + 2 sq. unit
5 –5 (D) e5 – e–5 – 2 sq. unit

20. Area enclosed between the curves |y| = 1  x2 and x2 + y2=1 is

3  8 8 2  8
(A) (B) (C) (D) none of these
3 3 3

21. The area of the closed figure bounded by x =  1, y = 0, y = x2 + x + 1 and the tangent to the curve y = x2 + x +
1 at A(1, 3) is
(A) 4/3 sq. units (B) 7/3 sq. units (C) 7/6 sq. units (D) none of these

22. The area of the region whose boundaries are defined by the curves y=2 cosx, y=3 tanx and the y-axis is
 2  3
(A) 1  3ln   sq. units (B) 1  ln 3  3 ln 2 sq. units
 3 2
3
(C) 1  ln 3  ln 2 (D) none of these
2

23. The area common to y2 = x and x2 = y is

(A) 1 (B) 2/3 (C) 1/3 (D) none of these

24. The area bounded by the curve y2 = 9x and the lines x = 1, x = 4 and y = 0, in the first quadrant, is
(A) 7 (B) 14 (C) 28 (D) 14/3
Corporate Branch Office : Noida Sec-62 Branch : C-56/33, Institutional Area, Sector-62, Noida
25. The area enclosed by y = 1 and  2x + y = 2 is (in square unit)
(A) 1/2 (B) 1/4 (C) 1 (D) none of these

26. The area of the region bounded by x2 = y, y = x + 2 and the x-axis is

(A) 8 (B) 4 (C) 2 (D) none of these

27. The area enclosed by y = ln x, its normal at (1, 0) and the y-axis is
(A) 1/2 (B) 3/2 (C) not defined (D) none of these

x2 y2
28. Area bounded by ellipse   1 is
a2 b 2
(A)  ab (B)  a2b (C)  ab2 (D) ab

29. Area bounded by y2 =  4x and its latus rectum is

4 8 2 16
(A) (B) (C) (D)
3 3 3 3

MULTI CHOICE MULTI CORRECT

1. If the area bounded by the curve y = x  x2 and line y = mx is equal to 9/2 sq. units, then m may be
(A)  4 (B)  2 (C) 2 (D) 4

2. If the area bounded by the curve y = sin 2x and lines x = /6, x = a and x-axis is equal to 1/2, then a is
(A) /3 (B) /2 (C)  /6 (D) 

x2
3. If the parabola y = divides the circle x2 + y2 = 8 into two parts, then the area of the parts may be
2
4 4 4 4
(A) 6  sq. units (B) 2  sq. units (C)   sq. units (D) 6  sq. units
3 3 3 3
NUMERICAL BASED TYPE
 1
1. The area enclosed between the curves y = log (x + e); x = loge   and x-axis is ____
y
2. Let F1, F2 are the foci of the ellipse 4x2 + 9y2 = 36 and P is a point on ellipse such that
PF1 = 2 PF2, then the area of triangle PF1F2 is ____

3. Area enclosed by  x   y   2 is _____

4. Area enclosed by y = g(x), x = – 3, x = 5 and x–axis where g(x) is inverse function of

f(x) = x3 + 3x + 1 is A, then [A] is __________. (where [.] denotes the greatest integer function

5. The area of the region bounded by y = |x – 1| and y = 1 is

 2 x ; x  0
6. Let f(x)   , then the area bounded by y = f(x) and y = 0 is
2  ; x  0
x

NUMERICAL BASED DECIMAL TYPE

1. The area bounded by y2 = 2x + 1 and x – y – 1 = 0 is _____

2. Area bounded by the curve [|x|] + [|y|] = 3, where [.] denotes the greatest integer function ___

3. The total area enclosed by the lines y = |x|, |x| = 1, and y = 0 is ____

Corporate Branch Office : Noida Sec-62 Branch : C-56/33, Institutional Area, Sector-62, Noida
4. The region M consists of all the points (x, y) satisfying the inequalities y  0, y  x and y  2 – x. The region N
which varies with the parameter t consists of all points which satisfy t  x  t + 1
 t  [0, 1], then min(ar(M  N)) is (in sq. units)

5. The area of the region bounded by the curve y = 2x – x2 and the line y = x is

6. The area bounded by the curve x = 6y – y2 and the y-axis is (in sq. units)

7. Let the curve, C: y = x6 + ax5 + bx4 + cx3 + dx2 + ex + f touches the line L : y = mx + n at x = 1, 2, 3. The area
bounded by these graph is
LINKED COMPREHENSION TYPE

Read the following write up carefully and answer the following questions:
Let O(0, 0), A(2, 0) be the vertices of an isosceles triangle inscribed in an ellipse (x  1)2 + 3y2 = 1. Let S represents
the region consisting all those points P inside the given triangle which states that, “Distance of point P from OA is not
more than the minimum distance of point P from the other two sides of the triangle”.

1. The coordinates of third vertex of the triangle is/are given by

(A) (1, 2) 
(B) 1, 3 
 1 
(C)  1,  (D) none of these
 3

2. If the third vertex of the triangle lies in the 1st quadrant, then the region S is represented by

(A) (B)
O A O A

(C) O A
(D)
O A

3. The area represented by the region S is

(A) 2  3 (B) 2  3
(C) 2  3 (D) 2  3

Read the following write up carefully and answer the following questions:

Consider function f(x) = 4  x 2 , g(x) = |x – 2| and h(x) = x  2 , for x  R a function is defined as F(x) = max or
min{f(x), g(x), h(x)} then
4. Area of F(x) = min{f(x), g(x), h(x)} between the co-ordinate axes for x < 0 is
(A) 2 sq. units (B)  sq. units
(C) 4 sq. units (D) none of these
5. Are enclosed by F(x) = min{f(x), g(x), h(x)} and G(x) = max{f(x), g(x), h(x)} for x  [0, 2] is
(A)  sq. units (B) ( + 2) sq. units
(C) ( – 2) sq. units (D) ( – 1) sq. units

Corporate Branch Office : Noida Sec-62 Branch : C-56/33, Institutional Area, Sector-62, Noida
 MATCH LIST TYPE
This section contains 1 multiple choice question. This question has matching lists. The codes for the lists have
choices (A), (B), (C) and (D) out of which ONLY ONE is correct.
1. Match the following:
List – I List – II
P. Area bounded by |2x + y| + |x  2y|  4 is 1. 1
Q. Area bounded by y = min (x  [x],  x  [ x] and x-axis (where x 
2.
 [0, 4]) [.] denotes the greatest integer function 2
R. Area bounded by |y| + 1/2  e|x| is 3. 32
S. Area bounded by y  sin sin1 x and x2 + y2 = 1 is 4. 2 (1  ln 2)
Codes :
P Q R S
(A) 3 2 1 4
(B) 3 1 4 2
(C) 3 1 2 4
(D) 4 1 2 3

2. A1 = Maximum area of quadrilateral whose two vertices lies on x-axis (between the points
(2, 0) and (2, 0) both inclusive) and two vertices on circumference of curve y  4  x . 2

a 4x 2
A2 = a + b where , ratio of area of region bounded by curve y = tanx and curve 2  y 2  1 , when a, b are
b 
prime numbers.
A3 = Sum of areas bounded by y = e x , x = 0, x = 1, y-axis and y = e x , x = 0, x = 1 and
3 2

y-axis.
A4 = is area bounded by y = f(x) from x = 1, to x = 3, where f(x) is satisfied the equation
1

  x  f  x  f  x  dx  12 .
1
0

LIST–I LIST–II
P. Minimum value of [A2]  [A1], [.] is G.I.F. 1. 6
Q. Minimum value of [A2 + A4], [.] is G.I.F. 2. 3
R. Value of [A3] + [A4] is is G.I.F. 3. 5
S. Value of [A1] + [A3] is is G.I.F. 4. 2
5. 4
6. 1
The correct option is :
(A) P  1; Q  3; R  4; S  3 (B) P  6; Q  1; R  2; S  5
(C) P  2; Q  4; R  4; S  1 (D) P  4; Q  1; R  3; S  5

MATCH FOLLOWING TYPE

Each question contains statements given in two columns which have to be matched. Statements (A, B, C, D) in
column I have to be matched with statements (p, q, r, s, t) in column II.
1. Match the following:
Column – I Column – II
(A) The area enclosed between the curves |x| + |y| = 2 and x2 = y in sq. units is 24
(p)
5
7 7
(B) The maximum value of the function f(x) = 3 sinx 4 cosx –will be given by (q)
3 3
(C) The length of common chord of two circles of radii 3 and 4 units which 16
intersect orthogonally is (r)
3
(D) The length of chord intercepted by the parabola y2 = 4(x + 1) passing 8
through its focus and inclined at 60° with positive x-axis is (s)
3
37
(t)
3

Corporate Branch Office : Noida Sec-62 Branch : C-56/33, Institutional Area, Sector-62, Noida
 MATCHING TYPE

Answer questions 1, 2 and 3 by appropriately matching the information given in the three columns of the following
table.

Column-I & Column-III describe certain regions in the x–y plane while Column-II gives the value of the area
Column–I Column–II Column–III
(I) The area enclosed by
(i) 27 (P) Area enclosed by y = 2x – x2 and y + 3 = 0
parabolas y2 = 4x and x2 = 8y
(II) Area enclosed by x2 + y2  1
(ii) 2 (Q) Area enclosed by |x| +|y|  
and y  x (x2 – 16).
(R) Area bounded by
2
(III) Area cut off from 4y = 3x2 by
(iii)
32 y   cos1  sin x   sin1  cos x   and x-axis
3 
2y = 3x + 12 3
between  x  2 , were [.] denotes G.I.F.
2
(S) y = f(x) and y = g(x) are two continuous
functions whose graphs intersect at 3 points
(0,4), (2,2) & (4,0) with f(x) > g(x) for x (0,2)

(IV) Area enclosed by curve (iv) and f(x) < g(x) for x  (2,4).
2 2
y 1 
4

 

 2  sin x   4 x  x
2
 If   f  x   g  x dx  7 and
0
4

  g  x   f  x dx  10
2

then area bounded between f(x) and g(x).

1. Which of the following options is the only CORRECT combination?
(A) (I) (ii) (P) (B) (I) (iii) (R)
(C) (I) (iv) (S) (D) (I) (iii) (P)

2. Which of the following options is the only CORRECT combination?

(A) (II) (iv) (S) (B) (II) (iv) (R)
(C) (II) (ii) (S) (D) (II) (ii) (R)

3. Which of the following options is the only CORRECT combination?

(A) (IV) (ii) (Q) (B) (III) (iv) (R)
(C) (III) (i) (Q) (D) (IV) (i) (S)

Corporate Branch Office : Noida Sec-62 Branch : C-56/33, Institutional Area, Sector-62, Noida
 ANSWERS
LPP - AREA
1 3 
1. 23/6 sq. units 2.  
8 8 6
7
3. 4. 7/3sq. units
6
5. (ii) b=1 7. a=9
8. 4 9. 1:1
10. f (x) = x3 – x2 11. b = 1/8, Aminimum = 4 3 sq. units
2
4a
12. 14. 64 sq. units
3
15. e 3 log2 sq. units
  1 158
16. (a) 2   , (b)
2 3 3
 5  4
17.  12  2  3  sq. 18. 
  3
19. 2 20. 8
MULTI CHOICE SINGLE CORRECT
1. B 2. C 3. C 4. B
5. A 6. A 7. D 8. A
9. A 10. C 11. C 12. B
13. D 14. A 15. D 16. A
17. A 18. A 19. B 20. A
21. C 22. B 23. C 24. B
25. A 26. D 27. B 28. A
29. B

MULTI CHOICE MULTI CORRECT

1. B, D 2. A, C 3. A, B

NUMERICAL BASED TYPE

1. 2 2. 4 3. 8 4. 4

5. 1 6. 2

NUMERICAL BASED DECIMAL TYPE

1. 5.3333 2. 16.00 3. 1.00 4. 0.75
5. 0.1666 6. 36.00 7. 0.1523

LINKED COMPREHENSION TYPE

1. C 2. C 3. A 4. B
5. C

Corporate Branch Office : Noida Sec-62 Branch : C-56/33, Institutional Area, Sector-62, Noida
 MATCH LIST TYPE

1. B 2. B

MATCH FOLLOWING TYPE

1. (A)  (q) (B)  (s) (C)  (p) (D)  (r)

MATCHING TYPE

1. D 2. B 3. A

Corporate Branch Office : Noida Sec-62 Branch : C-56/33, Institutional Area, Sector-62, Noida