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Application of Integrals · Question 15

JEE Main 2026 (Online) 23rd January Evening ShiftMCQMedium📈 Topic in 74% papers+4−1
1

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)

Answer

(1) 23(4 - 3 3)

Solution

Both circles have radius 2; centres (0, 0) and (0, 2).

Points of intersection: x2 + y2 = x2 + (y - 2)2 y = 1, x = 3

The common region is symmetric about y = 1, so

Area = 2 - 3 3 ( 4 - x2 - 1 )dx = 4 0 3 ( 4 - x2 - 1 )dx

0 3 4 - x2 dx = [ x2 4 - x2 + 2 -1 x2 ]0 3 = 32 + 2 3

Area = 4 ( 32 + 2 3 - 3 ) = 8 3 - 2 3

= 23(4 - 3 3)

Answer: (1)

JEE Main 2026 (Online) 23rd January Evening Shift

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2

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
JEE Main 2026 (Online) 22nd January Morning Shift
3

The area of the region A = \(x, y) : 4x2 + y2 8 and y2 4x\ is:

  1. + 23
  2. 2 + 2
  3. + 4
  4. 2 + 13
JEE Main 2026 (Online) 22nd January Evening Shift
4

The area of the region, inside the ellipse x2 + 4y2 = 4 and outside the region bounded by the curves y = |x| - 1 and y = 1 - |x|, is :

  1. 2 - 1
  2. 3( - 1)
  3. 2( - 1)
  4. 2 - 12
JEE Main 2026 (Online) 21st January Morning Shift
5

If the area of the region \(x, y) : 1 - 2x y 4 - x2, x 0, y 0\ is , , N, ( , ) = 1, then the value of ( + ) is:

  1. 73
  2. 85
  3. 67
  4. 91
JEE Main 2026 (Online) 21st January Evening Shift
6

Let the area of the bounded region \(x, y) : 0 9x y2, y 3x - 6 \ be A. Then 6A is equal to .

JEE Main 2025 (Online) 8th April Evening Shift
7

If the area of the region bounded by the curves y = 4 - x24 and y = x-42 is equal to , then 6 equals

  1. 210
  2. 250
  3. 240
  4. 220
JEE Main 2025 (Online) 7th April Morning Shift
8

If the area of the region \(x, y) : 1 + x2 y \x + 7, 11 - 3x\\ is A, then 3A is equal to :

  1. 50
  2. 46
  3. 49
  4. 47
JEE Main 2025 (Online) 7th April Evening Shift
9

If the area of the region \(x, y) : |x - 5| y 4 x\ is A, then 3A is equal to .

JEE Main 2025 (Online) 4th April Morning Shift
10

A line passing through the point A(-2, 0) touches the parabola P: y2 = x - 2 at the point B in the first quadrant. The area of the region bounded by the line AB, parabola P and the x-axis, is :

  1. 3
  2. 73
  3. 83
  4. 2
JEE Main 2025 (Online) 4th April Evening Shift
11

The area of the region bounded by the curve y = \|x|, x|x-2|\, the x-axis and the lines x = -2 and x = 4 is equal to

JEE Main 2025 (Online) 3rd April Morning Shift

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