2022 ACJC J2 MYE Q2

If \(1 < a < b\), solve \(\displaystyle\displaystyle \displaystyle\int_{0}^{b}{\left| a-{{x}^{2}} \right|}\mathrm{ d}x\), leaving your answer in terms of \(a\) and \(b\).[3]

Without the use of a graphic calculator, find the largest integer value of \(n\) such that \(\displaystyle\displaystyle \displaystyle\int_{\frac{n}{\mathrm{e}}}^{{{\mathrm{e}}^{\mathrm{e}}}}{\frac{1}{x{{\left( \ln x \right)}^{2}}}}\mathrm{ d}x\), where \(1 < \mathrm{e} < n\), is positive.[4]