2003 AMC 10B Problem 7

The symbolism \lfloor x \rfloor denotes the largest integer not exceeding x. For example, \lfloor 3 \rfloor = 3, and \lfloor 9/2 \rfloor = 4. Compute \lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{16} \rfloor.

\textbf{(A) } 35 \qquad\textbf{(B) } 38 \qquad\textbf{(C) } 40 \qquad\textbf{(D) } 42 \qquad\textbf{(E) } 136

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.