7 The diagram shows part of the curve $y = x ^ { 2 }$ for $0 \leqslant x \leqslant p$, where $p$ is a constant.\\
\includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-5_736_543_669_762}

The area $A$ of the region enclosed by the curve, the $x$-axis and the line $x = p$ is given approximately by the sum $S$ of the areas of $n$ rectangles, each of width $h$, where $h$ is small and $n h = p$. The first three such rectangles are shown in the diagram.\\
(i) Find an expression for $S$ in terms of $n$ and $h$.\\
(ii) Use the identity $\sum _ { r = 1 } ^ { n } r ^ { 2 } \equiv \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$ to show that $S = \frac { 1 } { 6 } p ( p + h ) ( 2 p + h )$.\\
(iii) Show how to use this result to find $A$ in terms of $p$.

\hfill \mbox{\textit{OCR H240/02 2018 Q7 [7]}}