AREA UNDER THE CURVE

LEVEL # 1
Questions
based on Area bounded by a curve

Q.1 The area between the curves y = 6 – x – x2 and x-axis is -

(A) 125/6 (B) 125/2
(C) 25/6 (D) 25/2

Q.2 The area between the curve y =ex and x-axis which lies between x = – 1 and
x = 1 is-
(A) e2 – 1 (B) (e2 –1)/e
(C) (1–e)/e (D) ( e– 1)/e2

Q.3 The area bounded by the curve y = sin 2x, x- axis and the ordinate x = /4 is-
(A) /4 (B) /2
(C) 1 (D) 1/2

Q.4 The area between the curve xy = a2, x-axis, x = a and x = 2a is-
(A) a log 2 (B) a2 log 2
(C) 2a log 2 (D) None of these


Q.5 Area under the curve y = sin 2x + cos 2x between x = 0 and x = , is-
4
(A) 2 sq. units (B) 1 sq. units
(C) 3 sq. units (D) 4 sq. units

Q.6 The area bounded by the curve y = 4x2 ; x = 0, y = 1 and y = 4 in the first quadrant is-
2 1
(A) 2 (B) 3
3 3
1 1
(C) 2 (D) 3
3 2
Q.7 The area between the curve y = sec x and y-axis when 1  y  2 is-
2 2
(A) – log ( 2 + 3 ) (B) + log ( 2 + 3)
3 3
 1
(C) – log (2 + 3 ) (D) None of these
3 2
Q.8 The area bounded by the lines y = x, y = 0 and x = 2 is-
(A) 1 (B) 2
(C) 4 (D) None of these

Q.9 The area bounded by the curve y = 1 + 8/x2, x-axis, x = 2 and x = 4 is-
(A) 2 (B) 3
(C) 4 (D) 5
Q.10 The area between the curve y = log x and x-axis which lies between x = a and x = b
(a > 1, b > 1) is-
(A) b log (b/e) – a log (a/e) (B) b log (b/e) + a log (a/e)
(C) log ab (D) log (b/a)
2
x
Q.11 Area bounded by the curve y = xe , x- axis and the ordinates x = 0, x =  is-
2 2
e  1 e  1
(A) sq. units (B) sq.units
2 2
2 2
 
(C) e  1 sq. units (D) e  1sq.units

Q.12 The area bounded between the curve y = 2x2 + 5, x-axis and ordinates x = – 2 and x = 1 is-
(A) 21 (B) 29/5
(C) 23 (D) 24

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Q.13 Area bounded by curve xy = c, x-axis between x = 1 and x = 4, is-
(A) c log 3 sq. units (B) 2 log c sq. units
(C) 2c log 2 sq. units (D) 2c log 5 sq. units


Q.14 The area bounded by the curve y = x sin x 2, x-axis and x = 0 and x = is-
2
(A) 1/2 (B) 1/ 2
(C) 1/4 (D) /2

x y
Q.15 The area bounded between the curve – + 1 = 0, x = – 2, x = 3 and x-axis is-
4 2
(A) 45/4 (B) 45/2
(C) 15 (D) 25/2

Q.16 The area bounded by curves y = tan x, x-axis and x = is-
3
(A) 2 log 2 (B) log 2
 2 
(C) log   (D) 0
 3

Q.17 The area between the curve x2 = 4ay, x-axis, and ordinate x = d is-
(A) d3 /12a (B) d3/a
(C) d3/2a (D) d3/6a
Q.18 Area bounded by the curve y = x (x – 1)2 and x-axis is-
(A) 4 (B) 1/3
(C) 1/12 (D) 1/2

Q.19 The area bounded by the curve y = loge x, x-axis and ordinate x = e is-
(A) loge2 (B) 1/2 unit
(C) 1 unit (D) e unit
1
Q.20 The area bounded by the curve y = , coordinates axes and x = /4 is-
cos2 x
(A) 1 (B) 2
(C) /4 (D) 

Q.21 The area between the curve y2 = 4x, y-axis, and y = – 1 and y = 3 is-
(A) 7/3 (B) 9/4
(C) 1/12 (D) 1/4

Q.22 The area bounded by the curve y = sin 2x, y-axis and the abscissa y = 1 is-
(A) 1 (B) 1/4
(C) /4 (D) (/4) – (1/2)

Q.23 The area between the curve x = 2y – y2 and y-axis is -

(A) 9/4 (B) 4/3
(C) 9 (D) None of these

Q.24 The area bounded by the curve x2 = 8y, x-axis and the ordinate x = – 2, x = 4 is-
(A) 4 (B) 2
(C) 1 (D) 3

Q.25 The area bounded by the curve y2 = x, straight line y = 4, and y-axis is-
(A) 16/3 (B) 64/3
(C) 7 2 (D) None of these

Q.26 The area between the curve y = sin3 x, x-axis, and the ordinates x = 0 to x = /2 is-
(A) 1 (B) 1/3
(C) 2/3 (D) 3/2

Q.27 The value of a for which the area of the region bounded by the curve y = sin 2x, the straight lines x =
/6, x = a and x-axis is equal to 1/2 is-
(A) /2 (B) /3

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 (C) 4/3 (D) /6

Q.28 The area of a loop bounded by the curve y = a sin x and x-axis is-
(A) a (B) 2a2
(C) 0 (D) 2a

Q.29 The area between the curves x = 2 – y – y2 and y-axis is-

(A) 9 (B) 9/2
(C) 9/4 (D) 3

Q.30 The area bounded by y = 4x – x2 and the x-axis is-

(A) 30/7 (B) 31/7
(C) 32/3 (D) 34/3

Q.31 The area contained between the x-axis and one arc of the curve y = cos 3x is-
(A) 1/3 (B) 2/3
(C) 2/7 (D) 2/5

Questions
based on Symmetric area

x2 y2
Q.32 The whole area of the ellipse  = 1 is-
a2 b2
(A) a2 b2 (B) a2b
(C) ab (D) ab2

Q.33 The area bounded by the curves y = 4x2 and y = 4 is-

(A) 7/3 (B) 14/3
(C) 5/3 (D) 16/3

Q.34 The area bounded between the curve |y| = 1 – x2 is-

(A) 2/3 (B) 4/3
(C) 8/3 (D) None of these

Q.35 The area bounded by the parabola y2 = 4ax, x-axis and x = 2a and latus rectum is-
(A) 2a2  
2 1 
(B) 4a2 2 2  1 
2
(C)
4 2
3
a 2 2 1   (D)
8a
3

2 2 1 
Q.36 The whole area bounded by the curves x = a cos t, y = b sin t is-

(A) ab (B)   ab
2

(C)   ab (D) None of these
4
Q.37 The whole area of the curve 9x2+16y2=144 is-
(A) 24 (B) 144
(C) 6 (D) 12

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Questions
based on Positive and negative area

Q.38 The area bounded by the curve y = sin x, x-axis and the lines x = 0 and x =  is-
(A) 1 (B) 3
(C) 2 (D) 4

Q.39 The area between the curve y = tan x and x-axis, when – /4  x  /4 is-
(A) log 2 (B) log 4
(C) log 2 (D) None of these

Q.40 The area bounded by the curve y = x3, the x-axis and the ordinates x = – 2 and x = 1 is-
(A) – 9 (B) 17/4
(C) – 15/4 (D) 15/4

Q.41 The area between the curve y = cos x and x- axis when /2  x  2 will be-
(A) 1 (B) 2
(C) 3 (D) 4

Q.42 The area between the curve y = cos x, x- axis, and the ordinates x = 0 and x = 2 is-
(A) 0 (B) 1
(C) 3 (D) 4
Questions
based on Area between two curves

Q.43 The area between the curves y2 = 4x and y = 2x is-

(A) 1/4 unit (B) 1/3 unit
(C) 1/2 unit (D) 2/3 unit

Q.44 The area bounded by the curvesy=ex,y=e–x and the line x = 1 is-
(A) e + e–1 (B) e + e–1 – 1
(C) e – e–1 + 1 (D) e + e–1 – 2

Q.45 The area bounded by the curve y = (x – 1) (x – 2) and coordinate axes is-
(A) 1/6 (B) 5/6
(C) 1/3 (D) 2/3

Q.46 The area bounded by the lines y = 2 + x and y = 2 – x and x-axis is-
(A) 3 (B) 4
(C) 8 (D) 16

Q.47 The area bounded by the curves y = sin x, y = cos x and x-axis from x= 0 to x=/2 is-
(A) 2 + 2 (B) 2
(C) 2 (D) 2– 2
Q.48 The area bounded between parabola x2 = 4y and y = |x| is -
(A) 2/3 (B) 4/3
(C) 8/3 (D) 16/3

Q.49 The area bounded by the curves y = x2 and y = |x| is-

(A) 2/3 (B) 1/6
(C) 1 (D) 1/3

Q.50 The common area of the curves y = x and x = y is-

(A) 3 (B) 5/3
(C) 1/3 (D) None of these

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Q.51 Area of the region bounded by the curves y = e x , y = e–x and the straight line y = 2 is-
(A) log (4/e) (B) 2 log (4/e)
(C) 4 log (4/e) (D) None of these

Q.52 The area bounded by y = tan x, y = cot x, x-axis in 0  x  is -
2
(A) log 2 (B) 3 log 2
(C) 2 log 2 (D) 4 log 2

Q.53 The area bounded by the curve y = 2x – x2 and straight line y = – x is -

(A) 35/6 (B) 9/2
(C) 43/6 (D) None of these
x2 y2 x y
Q.54 The area between the curve 2
 2 = 1 and the straight line  = 1 is-
a b a b
1 1 1
(A)  ab – ab (B) ab
4 2 4
1
(C)  ab (D) None of these
2
Q.55 The area of the figure bounded by the parabola y = x 2 + 1 and the straight line
x + y = 3 is-
(A) 3/2 (B) 5/2
(C) 7/2 (D) 9/2
Q.56 Common area between the parabolas y = 2x2 and y = x2 + 4 is-
(A) 16/3 (B) 8/3
(C) 32/3 (D) None of these

Q.57 If A is the area between the curve y = sin x and x-axis in the interval [0, /2], then the area between y
= sin 2x and x- axis in this interval will be-
(A) A (B) 2A
(C) A/2 (D) None of these
Q.58 Find the area enclosed by the lines y = x/2, y = 2x and x = 4 is-
(A) 1 (B) 2

(C) 12 (D) 16

LEVEL # 2
Q.1 The area bounded by the curve y = (x – 1)(x – 2) (x – 3), x-axis and ordinates x = 0,x = 3 is-
(A) 9/4 (B) 11/4
(C) 11/2 (D) None of these

Q.2 Area bounded by the curves y = 2x, y = 2x – x2, x = 0 and x = 2 is-

3 4 3 4
(A)  (B) 
log 2 3 log 2 3
4 4
(C) 3 log 2 + (D) 3 log 2 –
3 3

Q.3 Area bounded by the curves y = |x – 1|, y = 0 and |x| = 2 is-

(A) 5 (B) 4 (C) 9/2 (D) None of these

Q.4 The area of the region bounded by y2 = x and x = 36 is divided in the ratio 1: 7 by the line x = a, then a
equals-
(A) 7 (B) 8 (C) 9 (D) 0

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Q.5 The area of the triangle formed by the axes and a tangent to xy = a2 at (x1, yy,) on it is
a2 x1 a2 y 2
(A) (B) (C) 2a2 (D) 4a2
y1 x1

Q.6 The area bounded by the curve y = 1 – |x| and x-axis is -

(A) 1 (B) 1/2 (C) 2 (D) 1/3

Q.7 The area bounded between the curve x2 + y2 = 9 and lines y = 3/2, y = 3 and x = 0 is-
3 3 3 3
(A) (8+ 3 3 ) (B) (8– 3 3 ) (C) (4– 3 3 ) (D) (4– 3 3 )
4 4 2 8

Q.8 The area bounded by the curve y = f(x), x-axis and the ordinates x = 1 and x = b is (b – 1) sin (3b + 4)
, then f(x) equals-
(A) (x – 1) cos ( 3x + 4) (B) sin (3x + 4)
(C) sin (3x + 4) + 3 (x – 1) cos (3x + 4) (D) None of these

Q.9 The area bounded by the parabola x2 = 4y, the x-axis and the line x = 4 is divided into two equal area
by the line x = , then the value of  is-
(A) 21/3 (B) 22/3 (C) 24/3 (D) 25/3

Q.10 The area of the closed figure bounded by y = cos x, y = 1 + (2/) x and x = /2 is-
3  4 4
(A) 3/4 (B) /2 (C) (D)
4 4

 1
Q.11 The area enclosed between the curves y = loge (x + e), x = loge   and the x-axis is
y
(A) 1 (B) 2 (C) 3 (D) 4

Q.12 Area of the circle (x – 2)2 + (y – 3)2 = 32 which lies below the line y = x + 1 is-

 
6 6
   
  x  1  32   x  2  3 dx 32   x  2   3   x  1  dx
2 2
(A) (B)   
2 2
(C) 16  (D) None of these

LEVEL # 3
Q.1 The area bounded by the curve y = x |x|, x -axis and the ordinates x = 1, x = – 1 is given by -
1 2
(A) 0 (B) (C) (D) none of these.
3 3

Q.2 Area lying in the first quadrant and bounded by the circle x 2 + y2 = 4, the line x = 3 y and x-axis is -
 
(A)  (B) (C) (D) none of these.
2 3

Q.3 If area bounded by the curve y2 = 4 ax and y = mx is a2/3, then the value of m is -
(A) 2 (B) –2 (C) 1/2 (D) none of these.

Q.4 Area bounded by the curve y = x sinx and x-axis between x = 0 and x = 2 is
(A) 2 (B) 3 (C) 4 (D) none of these.

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

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

Q.6 The area of the figure bunded by the curves y = |x – 1| and y = 3 – |x| is -
(A) 2 (B) 3 (C) 4 (D) 1

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Assertion & Reason

(A) both S1 and S2 are true and S2 is the correct reason of S1

(B) both S1 and S2 are true and S2 is not the correct reason of S1
(C) S1 is true and S2 is false
(D) S1 is false and S2 is true

Q.7 Assertion(S1) : Area common to the curve y = x and x = y is 1/3 sq. units
b
Reason (S2) : Area =  a (g(x) – f(x)) dx.

y = g(x)

y = f(x)
x=b

Q.8 Assertion (S1) : Let the area bounded by the curve y = f(x), x-axis from x = 1 to x = a,a > 1 be 3a2 –
4a + 1 sq. units then f(x) = 6x – 4.
a
Reason (S2) : F(a) = Area =  f(x)dx  f(a)  F'(a)
1

Q.9 Assertion (S1) : the area bounded by the curves y = sin–1 x, y = cos–1 x and x-axis is 2 – 1.
1
2 1
 
 sin–1 xdx  1  2 – sin
–1
Reason (S2) : x  dx  2 – 1
0 
2

Passage
Consider the parabola y = x2 + 1 and the line x + y = 3
y
B

A
(0, 1)
  x

The line cuts the parabola at A and B. Let the abscissa of A and B be  and  .

Q.10 The values of  + and must be respectively

(A) 1 and 2 (B) 2 and 1
(C) –1 and –2 (D)None of these.

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Q.11 The area bounded between the parabola and the line in terms of  and  must be
(  )2 –    
(A) | – |  –2
3 2

2  2   
(B) | – |  2
3 2
(C) | – | | 2 – ( + ) – (2 + 2) |
(D) None of these.

Q.12 The area bounded between the parabola and the line must be
35
(A) 2 sq. units (B) sq. units
6
(C) 4 sq. units (D) None of these.

LEVEL # 4
[PREVIOUSLY ASKED QUESTIONS IN AIEEE & IIT]
Section - A

Q.1 If the area bounded by the x- axis, 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
[AIEEE 2002]
(A) (x  1) (B) (x  1)
x
(C) (x 2  1) (D)
1  x2
Q.2 The area of the region bounded by the curves y = | x – 1| and y = 3 – |x| is- [AIEEE 2003]
(A) 6 sq. units (B) 2 sq. units
(C) 3 sq. units (D) 4 sq. units

Q.3 The area of the region bounded by the curves y = |x – 2|, x = 1, x = 3 and the x- axis is-
(A) 1 (B) 2
(C) 3 (D) 4 [AIEEE 2004]

x2 y2
Q.4 Area of the greatest rectangle that can be inscribed in the ellipse + = 1 is - [AIEEE-2005]
a2 b2
(A) 2ab (B) ab
a
(C) ab (D)
b

Q.5 The area enclosed between the curve y = loge (x + e) and the coordinate axes is -
[AIEEE-2005]
(A) 1 (B) 2
(C) 3 (D) 4

Q.6 The parabolas y2 = 4x and x2 = 4y divide the square region bounded by the lines x = 4, y = 4 and the
coordinate axes. If S1, S2, S3 are respectively the areas of these parts numbered from top to bottom;
then S1 : S2 : S3 is - [AIEEE-2005]
(A) 1 : 2 : 1 (B) 1 : 2 : 3
(C) 2 : 1 : 2 (D) 1 : 1 : 1

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Q.7 Let f (x) be a non-negative continuous function such that the area bounded by the curve y = f (x), x-
     
axis and the ordinates x = and x =  > is   sin   cos   2  . Then f   is -
4 4  4  2
[AIEEE-2005]
         
(A)   2  1 (B)   2  1 (C)  1   2  (D)  1   2 
4  4   4   4 

Q.8 The area enclosed between the curves y2 = x and y = |x| is [AIEEE 2007]
(A) 2/3 (B) 1 (C) 1/6 (D) 1/3

Q.9 The area of the plane region bounded by the curves x + 2y2 = 0 and x + 3y2 = 1 is equal to -
[AIEEE 2008]
(A) 1/3 (B) 2/3 (C) 4/3 (D) 5/3

Q.10 The area of the region bounded by the parabola (y – 2)2 = x – 1, the tangent to the parabola at the
point (2, 3) and the x – axis is - [AIEEE 2009]
(A) 3 (B) 6 (C) 9 (D) 12

Section - B
Q.1 The area of the region bounded by y = |x–1| and y =1 is [IIT 93]
(A) 1 (B) 2 (C) 1/2 (D) None of these

Q.2 The slope of the tangent to the curve y = f(x) at a point (x,y) is 2x + 1 and the curve passes through
(1,2) . The area of the region bounded by the curve, the x- axis and the line x = 1 is
(A) 5/3 units (B) 5/6 units (C) 6/5 units (D) 6 units [IIT 95]

Q.3 The area bounded by the curves y = |x| – 1 and y = – |x| + 1 is- [IIT Scr.2002]
(A) 1 (B) 2 (C) 2 2 (D) 4

Q.4 Area of the region bounded by y = x , x= 2y + 3 & x- axis lying in 1st quadrant is-
[IIT Scr.2003]
(A) 2 3 (B) 18 (C) 9 (D) 34/3

Q.5 If area bounded by the curves x = ay2 and y = ax2 is 1, then a equals- [IIT Scr.2004]
1 1 1
(A) (B) (C) (D) 3
3 3 2

1
Q.6 Find the area between the curves y = (x – 1)2, y = (x + 1)2 and y = [IIT Scr.2005]
4
1 2 4 1
(A) (B) (C) (D)
3 3 3 6
Q.7 Area of the region bounded by the curve y = ex and lines x = 0 and y = e is -
[IIT 2009]
e 1 e
(A) e – 1 (B)  n(e  1– y)dy (C) e –  e x dx (D)  n y dy
1 0 1

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 ANSWER KEY
LEVEL # 1
Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Ans. A B D B B C A B C A B A C A A B A C C A
Q.No. 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Ans. A D B D B C B D B C B C D C C A D C A B
Q.No. 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58
Ans. C D B D B B D D D C B A B A D C A C

LEVEL # 2
Q.No. 1 2 3 4 5 6 7 8 9 10 11 12
Ans. B A A C C A D C D C B C

LEVEL # 3
Q.No. 1 2 3 4 5 6 7 8 9 10 11 12
Ans. C C A C A C A A A C A C

LEVEL # 4
Section – A
Q.No. 1 2 3 4 5 6 7 8 9 10
Ans. D D A A A D D C C C

Section – B
Q.No. 1 2 3 4 5 6 7
Ans. A B C C A A B,C,D

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