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Application of Integrals · Question 3

JEE Main 2026 (Online) 6th April Morning ShiftMCQMedium📈 Topic in 74% papers+4−1
1

Let e be the base of natural logarithm and let f : \1, 2, 3, 4\ \1, e, e2, e3 \ and g : \1, e, e2, e3 \ \1, 12, 13, 14 \ be two bijective functions such that f is strictly decreasing and g is strictly increasing. If (x)= [f-1 \g-1 ( 12 ) \ ]x, then the area of the region R= \(x, y) : x2 y (x), . . 0 x 1 \ is :

  1. 3- e(2)3 e(2)
  2. 13 e(2)
  3. 3+ e(2)
  4. 3+ e(2)2+ e(3)

Answer

(1) 3- e(2)3 e(2)

Solution

[figure]

f strictly decreasing : f(1)=e3, f(2)=e2, f(3)=e, f(4)=1

g strictly increasing, 14< 13< 12<1 : g(1)= 14, g(e)= 13, g (e2 )= 12, g (e3 )=1

g-1 ( 12 )=e2 and f-1 (e2 )=2

(x)=2x

For 0 x 1 : 2x 1 x2, so R lies between y=x2 (below) and y=2x (above)

Area = 01 (2x-x2 )dx= [ 2x e 2- x33 ]01

= 2-1 e 2- 13= 3- e(2)3 e(2)

Answer: (1)

JEE Main 2026 (Online) 6th April Morning Shift

Keep practising — Application of Integrals

2

The area of the region \(x, y) : x2-8x y -x \ is :

  1. 3436
  2. 6376
  3. 4376
  4. 5236
JEE Main 2026 (Online) 6th April Evening Shift
3

The area of the region R= \(x, y) : xy 27, 1 y x2 \ is equal to :

  1. 78 e 3- 523
  2. 54 e 3- 523
  3. 54 e 3- 263
  4. 54 e 3+ 263
JEE Main 2026 (Online) 5th April Morning Shift
4

Let f : R R be a function such that f(x)+3f ( 2-x )= x, x R. Let the maximum value of f on R be . If the area of the region bounded by the curves g(x)=x2 and h(x)= x3, >0, is 2, then 30 3 is equal to .

JEE Main 2026 (Online) 5th April Evening Shift
5

The area of the region \(x, y) : y -|x|, y |x x|, y 0\ is:

  1. 1+ 28
  2. 2+ 24
  3. 28-1
  4. 4+ 22
JEE Main 2026 (Online) 4th April Morning Shift
6

The area of the region bounded by the curves x+3y2=0 and x+4y2=1 is equal to :

  1. 13
  2. 23
  3. 43
  4. 53
JEE Main 2026 (Online) 4th April Evening Shift
7

If the area of the region bounded by 16x2-9y2=144 and 8x-3y=24 is A, then 3 (A+6 e(3) ) is equal to .

JEE Main 2026 (Online) 2nd April Evening Shift
8

The area of the region R = \(x, y) : xy 8, 1 y x2, x 0\ is

  1. 23(20 e(2) + 9)
  2. 13(40 e(2) + 27)
  3. 13(49 e(2) - 15)
  4. 23(24 e(2) - 7)
JEE Main 2026 (Online) 28th January Morning Shift
9

Let P1 : y = 4x2 and P2 : y = x2 + 27 be two parabolas. If the area of the bounded region enclosed between P1 and P2 is six times the area of the bounded region enclosed between the line y = x, > 0 and P1, then is equal to :

  1. 12
  2. 15
  3. 8
  4. 6
JEE Main 2026 (Online) 28th January Evening Shift
10

Let A1 be the bounded area enclosed by the curves y = x2 + 2, x + y = 8 and y-axis that lies in the first quadrant. Let A2 be the bounded area enclosed by the curves y = x2 + 2, y2 = x, x = 2, and y-axis that lies in the first quadrant. Then A1 - A2 is equal to

  1. 23(2 2 + 1)
  2. 23(3 2 + 1)
  3. 23( 2 + 1)
  4. 23(4 2 + 1)
JEE Main 2026 (Online) 24th January Morning Shift
11

Let f( ) denote the area of the region in the first quadrant bounded by x = 0, x = 1, y2 = x and y = | x - 5| - |1 - x| + x2. Then (f(0) + f(1)) is equal to

  1. 12
  2. 14
  3. 9
  4. 7
JEE Main 2026 (Online) 24th January Evening Shift

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