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Application of Integrals · Question 1

JEE Main 2026 (Online) 8th April Evening ShiftMCQMedium📈 Topic in 74% papers+4−1
1

Let f : (1, ) R be a function defined as f(x)= x-1x+1. Let fi+1(x)=f (fi(x) ), i=1, 2, , 25, where f1(x)=f(x). If g(x)+f26(x)=0, x (1, ), then the area of the region bounded by the curves y=g(x), 2y=2x-3, y=0 and x=4 is :

  1. 18+ e 2
  2. 14+ e 2
  3. 56+3 e 2
  4. 56+ e 2

Answer

(1) 18+ e 2

Solution

f2(x)=f(f(x))= x-1x+1-1 x-1x+1+1= -22x=- 1x

f4(x)=f2 (f2(x) )=- 1- 1x=x f4m+2=f2

26=4(6)+2 f26(x)=- 1x g(x)= 1x

2y=2x-3 meets y=0 at x= 32 and y= 1x where 2x2-3x-2=0 x=2 (as x>1), i.e. at (2, 12 )

[figure]

Area = 3/22 (x- 32 )dx+ 24 dxx

= 12 ( 12 )2+ e 42= 18+ e2

Answer: (1)

JEE Main 2026 (Online) 8th April Evening Shift

Keep practising — Application of Integrals

2

The area of the region \(x, y) : 0 y 6-x, . . y2 4x-3, x 0 \ is :

  1. 8
  2. 9
  3. 12
  4. 15
JEE Main 2026 (Online) 6th April Morning Shift
3

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)
JEE Main 2026 (Online) 6th April Morning Shift
4

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
5

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
6

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
7

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
8

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
9

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
10

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
11

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

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