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Application of Integrals

JEE Main 2026 (Online) 5th April Evening ShiftNumericalMedium📈 Topic in 72% papers+4
Ques. 1JEE Main 2026, 5 April (Evening Shift)

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 .

Answer

16

Solution

[figure]

Given f(x) + 3f ( 2 - x ) = x …(i); replacing x by 2 - x gives f ( 2 - x ) + 3f(x) = x …(ii)

Then 3 (ii) − (i): 8f(x) = 3 x - x, so f(x) = 3 x - x8 and

= 32 + 128 = 108 2 = 1064 = 532

x2 = x3 gives x = 0 or x = 1 , and for 0 < x < 1 , x2 > x3, so

Area = 01/ (x2 - x3 )dx = 13 3 - 14 3 = 112 3

112 3 = 532 3 = 3260 = 815

30 3 = 16

Answer: 16

Keep practising — Application of Integrals

Ques. 2JEE Main 2026, 4 April (Morning Shift)

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
Ques. 3JEE Main 2026, 4 April (Evening Shift)

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
Ques. 4JEE Main 2026, 2 April (Evening Shift)

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 .

Ques. 5JEE Main 2026, 28 January (Morning Shift)

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)
Ques. 6JEE Main 2026, 28 January (Evening Shift)

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
Ques. 7JEE Main 2026, 24 January (Morning Shift)

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)
Ques. 8JEE Main 2026, 24 January (Evening Shift)

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
Ques. 9JEE Main 2026, 23 January (Morning Shift)

Let the area of the region bounded by the curve y = \ x, x\, lines x = 0, x = 3 2, and the x-axis be A. Then, A + A2 is equal to .

Ques. 10JEE Main 2026, 23 January (Evening Shift)

The area of the region enclosed between the circles x2 + y2 = 4 and x2 + (y - 2)2 = 4 is:

  1. 23(4 - 3 3)
  2. 43(2 - 3)
  3. 43(2 - 3 3)
  4. 23(2 - 3 3)
Ques. 11JEE Main 2026, 22 January (Morning Shift)

Let the line x = -1 divide the area of the region \(x, y) : 1 + x2 y 3 - x \ in the ratio m : n, (m, n) = 1. Then m + n is equal to

  1. 27
  2. 28
  3. 25
  4. 26

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