Definite integral as the limit of a sum

Four rectangles stand on $[0, 1]$, each $\tfrac14$ wide and as tall as the line $y = x$ at its right end. Their total area is $\tfrac{1 + 2 + 3 + 4}{16} = \tfrac58$, but the triangle under the line has area $\tfrac12$. Can rectangles ever give exactly $\tfrac12$?