The diagram shows the curve with equation \(y=1-x^{2}\) 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 the rectangles, show that
\[\int_{0}^{1}\left(1-x^{2}\right) \mathrm{d} x<\frac{4 n^{2}+3 n-1}{6 n^{2}} .\]
(b) Use a similar method to find, in terms of \(n\), a lower bound for \(\int_{0}^{1}\left(1-x^{2}\right) \mathrm{d} x\).