Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If ∫₀ˣ √(1 - (f'(t))²) dt = ∫₀ˣ f(t)dt, 0 ≤ x ≤ 1 and f(0) = 0, then lim x → 0 1/(x²)∫₀ˣ f(t)dt:

Integral Calculus: JEE Main 2021 Mathematics question

JEE Main 2021 · 31 Aug, Morning shift · Q3313

JEE Main 202131 Aug, Morning shiftMathematicsMediumIntegral Calculus369× since 2002

Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If

\int_0^x {\sqrt {1 - {{(f'(t))}^2}} dt = \int_0^x {f(t)dt} }

0 \le x \le 1

and f(0) = 0, then

\mathop {\lim }\limits_{x \to 0} {1 \over {{x^2}}}\int_0^x {f(t)dt}

JEE Main 2021 Integral Calculus Question with Solution (31 Aug Morning shift, Q3313) | JEE Arc

JEE Main 2021 · 31 Aug, Morning shift · Q3313

JEE Main 202131 Aug, Morning shiftMathematicsMediumIntegral Calculus369× since 2002

Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If

\int_0^x {\sqrt {1 - {{(f'(t))}^2}} dt = \int_0^x {f(t)dt} }

0 \le x \le 1

and f(0) = 0, then

\mathop {\lim }\limits_{x \to 0} {1 \over {{x^2}}}\int_0^x {f(t)dt}