Let S_n denote the finite sum 1 + 2^2/3 + 3^2/3 + . . . + n^2/3.

1. Calculate the integral J = ∫(b=1000;a=0) x^2/3, expressing your answer as a

positive integer.

2. Use suitable upper and lower Riemann sums for the function f(x) = x^2/3

on the interval [0, 1000] to prove that S_999 < J < S_1000.

3. Hence, or otherwise, find integer lower and upper bounds, no more than

100 units apart, for S_1000.

I sat on this for 1 hour and could not solve. Please I need a clear explanation and as simple as you can so I can understand it. Any Help appreciated