The diagram shows the curve with equation \(y=x^{2}\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac{1}{n}\).
(a) By considering the sum of the areas of these rectangles, show that
\[\int_{0}^{1} x^{2} \mathrm{~d} x<\frac{2 n^{2}+3 n+1}{6 n^{2}} .\]
(b) Use a similar method to find, in terms of \(n\), a lower bound for \(\int_{0}^{1} x^{2} \mathrm{~d} x\).