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\includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-04_673_1285_1176_429} The diagram shows the curve with equation \(y = \sqrt { x }\). A set of \(N\) rectangles of unit width is drawn, starting at \(x = 1\) and ending at \(x = N + 1\), where \(N\) is an integer (see diagram).

1. By considering the areas of these rectangles, explain why $$\sqrt { 1 } + \sqrt { 2 } + \sqrt { 3 } + \ldots + \sqrt { N } < \int _ { 1 } ^ { N + 1 } \sqrt { x } \mathrm {~d} x$$
2. By considering the areas of another set of rectangles, explain why $$\sqrt { 1 } + \sqrt { 2 } + \sqrt { 3 } + \ldots + \sqrt { N } > \int _ { 0 } ^ { N } \sqrt { x } \mathrm {~d} x$$
3. Hence find, in terms of \(N\), limits between which \(\sum _ { r = 1 } ^ { N } \sqrt { r }\) lies. \section*{Jan 2006}