If a is a positive real number such that ₀^ a [x]^2 , dx = 85 2 , where [ ] denotes the greatest integer function, then the value of 4a is

Step-by-step solution

Let a = n + f , where n = [a] is a non-negative integer and f = a [0, 1) is the fractional part of a . The given integral can be split into integer intervals and a final fractional interval: ₀^ a [x]^2 , dx = ₀¹ 0^2 , dx + ₁² 1^2 , dx + + _ n-1 ^ n (n-1)^2 , dx + _ n ^ n+f n^2 , dx Evaluating the integrals, we get: _ k=0 ^ n-1 k^2 (1) + n^2 (f) = 85 2 (n-1)n(2n-1) 6 + n^2 f = 42.5 We test integer values for n to find where the remaining area can be satisfied by a valid fraction f [0, 1) . If n = 4 : The sum of squa