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JEE Main Definite Integrals and Areas questions

102 Mathematics questions on Definite Integrals and Areas as asked in JEE Main, newest first. 58 come with a worked solution.

  1. Let T and C respectively be the transverse and conjugate axes of the hyperbola 16x^2 - y^2 + 64x + 4y + 44 = 0. Then the area of the region above the parabola x^2 = y + 4, below the transverse axis T and on the right of…

    2023 · with solution

  2. Let A = \(x, y) ∈ R^2 : y ≥ 0, 2x ≤ y ≤ √(4-(x-1)^2)\ and B = \(x, y) ∈ R × R : 0 ≤ y ≤ min \2x, √(4-(x-1)^2) \\. Then the ratio of the area of A to the area of B is

    2023

  3. Let Δ be the area of the region \(x,y) ∈ R^2 : x^2 + y^2 ≤ 21, y^2 ≤ 4x, x ≥ 1 \. Then 1/2 (Δ - 21sin^-1frac2√(7) ) is equal to

    2023

  4. Let [x] denote the greatest integer ≤ x. Consider the function f(x) = max\x^2, 1+[x]\. Then the value of the integral int_0^2 f(x) dx is

    2023 · with solution

  5. If f(x) = dfrac2^2x2^2x+2, x ∈ R, then f (1/2023 ) + f (2/2023 ) + … + f (2022/2023 ) is equal to

    2023 · with solution

  6. The integral 16int_1^2 dx/(x^3(x^2+2)^2) is equal to

    2023

  7. int_frac3√(2)4^frac3√(3)4 frac48√(9-4x^2) dx is equal to

    2023 · with solution

  8. The area enclosed by the curves y^2 + 4x = 4 and y - 2x = 2 is :

    2023

  9. Let f(x) = x + a/(π^2 - 4)sin x + b/(π^2 - 4)cos x, x ∈ R be a function which satisfies f(x) = x + displaystyleint_0^π/2 sin(x+y) f(y) dy. Then (a+b) is equal to

    2023

  10. The minimum value of the function f(x) = int_0^2 e^|x-t| dt is :

    2023

  11. The value of 8/(π) int_0^(π)/2 dfrac(cos x)^2023(sin x)^2023 + (cos x)^2023 dx is __________

    2023 · with solution

  12. If int_1^e [log_e x] dx = m/n log_e ( (e^2)/c ), where m and n are coprime natural numbers, then m^2 + n^2 - 5 is equal to

    2023

  13. The value of 12int_0^3 |x^2 - 3x + 2 | dx is ___________

    2023

  14. If the area of the region bounded by the curves y^2 - 2y = -x, x + y = 0 is A, then 8A is equal to

    2023

  15. The value of int_-π/2^π/2 dx/([x] + [sin x] + 4), where [t] denotes the greatest integer less than or equal to t, is

    2019

  16. A value of α such that int_α^α+1 dx/((x+α)(x+α+1)) = log_e (9/8 ) is

    2019 · with solution

  17. lim_n → ∞ ( frac(n+1)^1/3n^4/3 + frac(n+2)^1/3n^4/3 + ..... + frac(2n)^1/3n^4/3 ) is equal to

    2019 · with solution

  18. If int_x^x f(t) dt = x^2 + int_x^1 t^2 f(t) dt, then f (1/2 ) is

    2019

  19. If the area (in sq units) bounded by the parabola y^2 = 4λ x and the line y = λ x, λ > 0, is 1/9, then λ is equal to

    2019 · with solution

  20. If the area enclosed between the curves y = kx^2 and x = ky^2, (k > 0), is 1 square unit. Then, k is

    2019

  21. lim_n → ∞ ( n/(n^2+1^2) + n/(n^2+2^2) + n/(n^2+3^2) + … + 1/5n ) is equal to

    2019

  22. If int_0^π/2 (cot x)/(cot x + cosec x) dx = m(π + n), then m · n is equal to

    2019

  23. If the area (in sq units) of the region \(x, y): y^2 ≤ 4x, x + y ≤ 1, x ≥ 0, y ≥ 0\ is a√(2) + b, then a - b is equal to

    2019

  24. The integral int_π/6^π/3 sec^2/3 x csc^4/3 x dx is equal to

    2019

  25. The area (in sq units) of the region bounded by the curves y = 2^x and y = |x+1| in the first quadrant is

    2019

  26. The value of ∫limits_0^2π [sin 2x(1 + cos 3x)] dx, where [t] denotes the greatest integer function, is

    2019

  27. The area (in sq units) of the region A = \(x, y): (y^2)/2 ≤ x ≤ y + 4 \ is

    2019 · with solution

  28. The value of the integral ∫limits_0^1 xcot^-1(1-x^2+x^4)dx is

    2019

  29. The area (in sq units) of the region A = \(x, y) : x^2 ≤ y ≤ x + 2\ is

    2019 · with solution

  30. The value of int_0^π/2 (sin^3 x)/(sin x + cos x) dx is

    2019

  31. Let S(α) = \(x, y) : y^2 ≤ x, 0 ≤ x ≤ α\ and A(α) is area of the region S(α). If for λ, 0 < λ < 4, A(λ) : A(4) = 2 : 5, then λ equals

    2019 · with solution

  32. Let f(x) = ∫limits_0^x g(t)dt, where g is a non-zero even function. If f(x+5) = g(x), then ∫limits_0^x f(t)dt equals

    2019

  33. If f(x) = (2 - xcos x)/(2 + xcos x) and g(x) = log_e x, (x > 0) then the value of the integral int_-π/4^π/4 g(f(x)) dx is

    2019 · with solution

  34. The area (in sq units) of the region A = \(x, y) ∈ R × R mid 0 ≤ x ≤ 3, 0 ≤ y ≤ 4, y ≤ x^2 + 3x\ is

    2019 · with solution

  35. Let f and g be continuous functions on [0, a] such that f(x) = f(a - x) and g(x) + g(a - x) = 4, then int_0^a f(x) g(x) dx is equal to

    2019

  36. The area (in sq units) of the region bounded by the parabola, y = x^2 + 2 and the lines, y = x + 1, x = 0 and x = 3, is

    2019

  37. If f(x) = ∫ (5x^8 + 7x^6)/((x^2 + 1 + 2x^7)^2) dx, \ (x ≥ 0), and f(0) = 0, then the value of f(1) is

    2019 · with solution

  38. The integral int_1^e [ (x/e )^2x - (e/x )^x ]log_e x dx is equal to

    2019

  39. The area (in sq units) in the first quadrant bounded by the parabola, y = x^2 + 1, the tangent to it at the point (2, 5) and the coordinate axes is

    2019 · with solution

  40. The integral int_π/6^π/4 dx/(sin 2x(tan^5 x + cot^5 x)) equals

    2019

  41. The area (in sq units) of the region bounded by the curve x^2 = 4y and the straight line x = 4y - 2 is

    2019 · with solution

  42. The value of the integral int_-2^2 (sin^2 x)/( [x/(π) ] + 1/2) dx (where, [x] denotes the greatest integer less than or equal to x) is

    2019 · with solution

  43. The area of the region A = \(x, y) 0 ≤ y ≤ |x| + 1 and -1 ≤ x ≤ 1\ in sq. units, is

    2019

  44. The area (in sq units) bounded by the parabola y = x^2 - 1, the tangent at the point (2, 3) to it and the Y-axis is

    2019

  45. If int_0^π/3 fractanθ√(2ksecθ) dθ = 1 - frac1√(2), (k > 0), then the value of k is

    2019 · with solution

  46. The value of int_-π^π |cos x|^3 dx is

    2019 · with solution

  47. Let g(x) = cos x^2, f(x) = √(x) and α, β (α < β) be the roots of the quadratic equation 18x^2 - 9π x + π^2 = 0. Then, the area (in sq units) bounded by the curve y = (gof)(x) and the lines x = α, x = β and y = 0, is

    2018 · with solution

  48. The area (in sq units) of the region \(x, y) : x ≥ 0, x + y ≤ 3, x^2 ≤ 4y and y ≤ 1 + √(x)\ is

    2017

  49. int_π/4^3π/4 dx/(1 + cos x) is equal to

    2017 · with solution

  50. The area (in sq units) of the region \(x, y): y^2 ≥ 2x and x^2 + y^2 ≤ 4x, x ≥ 0, y ≥ 0\ is

    2016 · with solution

  51. lim_n → ∞ ( frac(n+1)(n+2)·s 3nn^2n )^1/n is equal to

    2016 · with solution

  52. Area of the region \(x, y) ∈ R^2 : y ≥ √(|x+3|), 5y ≤ (x+9) ≤ 15\ is equal to

    2016

  53. The value of int_-π/2^π/2 (x^2 cos x)/(1 + e^x) dx is equal to

    2016 · with solution

  54. The area (in sq units) of region described by \(x, y) y^2 ≤ 2x and y ≥ 4x - 1\ is

    2015 · with solution

  55. The integral int_2^4 (log x^2)/(log x^2 + log(36 - 12x + x^2)) dx is equal to

    2015 · with solution

  56. Let f:[0,2] → R be a function which is continuous on [0,2] and is differentiable on (0,2) with f(0)=1. Let F(x) = int_0^x^2 f(√(t)) dt, for x ∈ [0,2]. If F'(x) = f'(x), forall x ∈ (0,2), then F(2) equals

    2014

  57. The common tangents to the circle x^2 + y^2 = 2 and the parabola y^2 = 8x touch the circle at the points P, Q and the parabola at the points R, S. Then, the area (in sq units) of the quadrilateral PQRS is

    2014

  58. The area (in sq units) of the region described by A = \(x, y) : x^2 + y^2 ≤ 1 and y^2 ≤ 1 - x\ is

    2014

  59. The area enclosed by the curves y = sin x + cos x and y = |cos x - sin x| over the interval [0, (π)/2 ] is

    2014 · with solution

  60. The integral int_π/4^π/2 (2cosec x)^17 dx is equal to

    2014 · with solution

  61. The integral ∫limits_0^(π)/2 √(1 + 4sin^2x/2 - 4sinx/2) dx is equal to

    2014

  62. The intercepts on X-axis made by tangents to the curve, y = int_0^x |t| dt, x ∈ R, which are parallel to the line y = 2x, are equal to

    2013

  63. Let f : [1/2, 1] → R (the set of all real numbers) be a positive, non-constant and differentiable function such that f'(x) < 2f(x) and f(1/2) = 1. Then, the value of int_1/2^1 f(x) dx lies in the interval

    2013 · with solution

  64. The area (in sq units) bounded by the curves y = √(x), 2y - x + 3 = 0, X-axis and lying in the first quadrant, is

    2013 · with solution

  65. The value of the integral int_-π/2^π/2 (x^2 + log(π - x)/(π + x) ) cos x dx is

    2012 · with solution

  66. Let f: [-1, 2] → [0, ∞) be a continuous function such that f(x) = f(1-x), forall x ∈ [-1, 2]. If R_1 = int_-1^2 x f(x) dx and R_2 are the area of the region bounded by y = f(x), x = -1, x = 2 and the x-axis. Then,

    2011

  67. If the straight line x = b divide the area enclosed by y = (1-x)^2, y = 0 and x = 0 into two parts R_1(0 ≤ x ≤ b) and R_2(b ≤ x ≤ 1) such that R_1 - R_2 = 1/4. Then, b equals

    2011

  68. The value of int_√(log 2)^√(log 3) (x sin x^2)/(sin x^2 + sin(log 6 - x^2)) dx is \hspace{1em}

    2011 · with solution

  69. The area of the region between the curves y = √((1+sin x)/(cos x)) and y = √((1-sin x)/(cos x)) and bounded by the lines x = 0 and x = (π)/4 is

    2008 · with solution

  70. The area of the equilateral triangle, in which three coins of radius 1 cm are placed, as shown in the figure, is [figure]

    2005 · with solution

  71. If int_sin x^1 t^2 f(t) dt = 1 - sin x, forall x ∈ (0, π/2), then f (frac1√(3) ) is

    2005 · with solution

  72. The value of int_-1^0 [x^3 + 3x^2 + 3x + 3 + (x+1)cos(x+1)] dx is

    2005

  73. The area bounded by the curves y = (x-1)^2, y = (x+1)^2 and y = 1/4 is

    2005 · with solution

  74. If f(x) is differentiable and int_0^t^2 x f(x) dx = 2/5t^5, then f (4/25 ) equals

    2004 · with solution

  75. The value of the integral int_0^1 √((1-x)/(1+x)) dx is

    2004 · with solution

  76. The area enclosed between the curves y = ax^2 and x = ay^2 (a > 0) is 1 sq unit. Then, the value of a is

    2004

  77. The area of the quadrilateral formed by the tangents at the end points of latusrectum to the ellipse (x^2)/9 + (y^2)/5 = 1, is

    2003 · with solution

  78. If f(x) = int_x^2^x^2+1 e^-t^2 dt, then f(x) increases in

    2003 · with solution

  79. If I(m,n) = int_0^1 t^m(1+t)^n dt, then the expression for I(m,n) in terms of I(m+1, n-1) is

    2003 · with solution

  80. Let T > 0 be a fixed real number. Suppose, f is a continuous function such that for all x ∈ R, f(x + T) = f(x). If I = int_0^T f(x) dx, then the value of int_3^3+3T f(2x) dx is

    2002 · with solution

  81. The area (in sq units) bounded by the curves y = |x| - 1 and y = -|x| + 1 is

    2002 · with solution

  82. Let f(x) = int_1^x √(2 - t^2) dt. Then, the real roots of the equation x^2 - f'(x) = 0 are

    2002 · with solution

  83. The integral int_-1/2^1/2 [ [x] + log ((1+x)/(1-x) ) ] dx equals

    2002 · with solution

  84. The triangle formed by the tangent to the curve f(x) = x^2 + bx - b at the point (1,1) and the coordinate axes, lies in the first quadrant. If its area is 2 sq units, then the value of b is

    2001

  85. Let f:(0,∞) → R and F(x) = int_0^x f(t) dt. If F(x^2) = x^2(1+x), then f(4) equals

    2001 · with solution

  86. The value of int_-π^π (cos^2 x)/(1 + a^x) dx, a > 0, is

    2001

  87. If f(x) = begincases e^cos x sin x & , for |x| ≤ 2 \ 2 & , otherwise endcases then int_-2^3 f(x) dx is equal to

    2000 · with solution

  88. The value of the integral int_e^-1^e^2 | (log_e x)/x | dx is

    2000 · with solution

  89. If for a real number y, [y] is the greatest integer less than or equal to y, then the value of the integral int_π/2^3π/2 [2sin x] dx is

    1999

  90. int_π/4^3π/4 dx/(1 + cos x) is equal to

    1999 · with solution

  91. If int_0^x f(t) dt = x + int_x^1 t f(t) dt, then the value of f(1) is

    1998 · with solution

  92. If g(x) = int_0^x cos^4 t dt, then g(x + π) equals

    1997 · with solution

  93. Let f be a positive function. If I_1 = int_1-k^k x f [x(1-x) ] dx and I_2 = int_1-k^k f [x(1-x) ] dx, where 2k - 1 > 0. Then, I_1/I_2 is

    1997 · with solution

  94. The value of int_0^2π [2sin x] dx, where [·] represents the greatest integral functions, is

    1995

  95. If f(x) = Asin ((π x)/2 ) + B, f' (1/2 ) = √(2) and int_0^1 f(x) dx = 2A/(π), then constants A and B are

    1995 · with solution

  96. The value of int_0^π/2 dx/(1+tan^3 x) is

    1993 · with solution

  97. Let f: R → R be a differentiable function and f(1) = 4. Then, the value of lim_x → 1 int_4^f(x) 2t/(x-1) dt is

    1990 · with solution

  98. Let f: R → R and g: R → R be continuous functions. Then, the value of the integral int_-π/2^π/2 [f(x) + f(-x)][g(x) - g(-x)] dx

    1990

  99. For any integer n, the integral int_0^π e^cos^2 x cos^3(2n+1)x dx has the value

    1985

  100. The value of the integral int_0^π/2 dfrac√(cot x)√(cot x) + √(tan x) dx is

    1983 · with solution

  101. The value of the definite integral int_0^1 (1 + e^-x^2) dx is

    1981 · with solution

  102. The slope of tangent to a curve y = f(x) at [x, f(x)] is 2x + 1. If the curve passes through the point (1, 2), then the area bounded by the curve, the X-axis and the line x = 1 is

    with solution

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