The pictorial proof of the sum of [superscript n][subscript k=1] k[superscript 2] = 1/6n(n+1)(2n+1) is represented in the form of an integral. The integral representations are also applicable to the sum of [superscript n][subscript k-1] k[superscript m] (m greater than or equal to 3). These representations reveal that the sum of [superscript n][subscript k=1] k[superscript m] can be solved into a factor n(n+1) (proportional to the sum of [superscript n][subscript k=1] k, where m is a positive integer. (Contains 4 figures.)