JEE Advanced Definite Integrals and Areas questions

7 Mathematics questions on Definite Integrals and Areas as asked in JEE Advanced, newest first.

Let S = \(x, y) ∈ R × R : x ≥ 0, y ≥ 0, y^2 ≤ 4x, y^2 ≤ 12 - 2x and 3y + √(8) x ≤ 5√(8) \. If the area of the region S is α√(2), then α is equal to
2024
Let the function f:[1,∞) → R be defined by f(t) = begincases (-1)^n+12, & if t = 2n-1, n ∈ N, \ ((2n+1-t))/2 f(2n-1) + ((t-(2n-1)))/2 f(2n+1), & if 2n-1 < t < 2n+1, n ∈ N. endcases Define g(x) = displaystyleint_1^x f(t)…
2024
Paragraph Let f : [0, (π)/2 ] → [0,1] be the function defined by f(x) = sin^2 x and let g : [0, (π)/2 ] → [0, ∞) be the function defined by g(x) = √((π x)/2 - x^2). The value of 2displaystyleint_0^(π)/2 f(x)g(x)dx -…
2024
Paragraph Let f : [0, (π)/2 ] → [0,1] be the function defined by f(x) = sin^2 x and let g : [0, (π)/2 ] → [0, ∞) be the function defined by g(x) = √((π x)/2 - x^2). The value of 16/(π^3) displaystyleint_0^(π)/2…
2024
Let f : [0,1] → [0,1] be the function defined by f(x) = (x^3)/3 - x^2 + 5/9x + 17/36. Consider the square region S = [0,1] × [0,1]. Let G = \(x, y) ∈ S : y > f(x)\ be called the green region and R = \(x, y) ∈ S : y <…
2023
For x ∈ R, let tan^-1(x) ∈ (-(π)/2, (π)/2 ). Then the minimum value of the function f : R → R defined by f(x) = displaystyle∫limits_0^xtan^-1x dfrace^(t-cos t)1+t^2023 dt is
2023
Let n ≥ 2 be a natural number and f:[0,1] → R be the function defined by f(x) = begincases n(1-2nx) & if 0 ≤ x ≤ 1/2n \ 2n(2nx-1) & if 1/2n ≤ x ≤ 3/4n \ 4n(1-nx) & if 3/4n ≤ x ≤ 1/n \ n/(n-1)(nx-1) & if 1/n ≤ x ≤ 1…
2023

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