Calculus question?

Please see the graph below,
Ak
= ( (k+1) - k ) * [ k ] , where [ ] denotes floor function
= 1 * k
= k

∫ [ t ] dt , from t = 0 to t = n
= Σ Ak , from k = 0 to k = n-1
= Σ k , from k = 0 to k = n-1
= 0 + 1 + 2 + ..... + (n-1)
= { 1 + (n-1) } (n-1)/2
= n(n-1)/2
= [n] ( [n] - 1 ) / 2 , since n is an integer

Q.E.D.