Explanation

(d)
$\int_{0}^{2}{\left\lbrack x^{2} \right\rbrack dx}$
$= \int_{0}^{1}{\left\lbrack x^{2} \right\rbrack dx + \int_{1}^{\sqrt{2}}{\left\lbrack x^{2} \right\rbrack dx = \int_{\sqrt{2}}^{\sqrt{3}}{\left\lbrack x^{2} \right\rbrack dx + \int_{\sqrt{3}}^{2}{\left\lbrack x^{2} \right\rbrack dx}}}}$
$= \int_{0}^{1}{0\ dx + \int_{1}^{\sqrt{2}}{1dx + \int_{\sqrt{2}}^{\sqrt{3}}{2\ dx + \int_{\sqrt{3}}^{2}{3dx}}}}$
$= \lbrack x\rbrack_{1}^{\sqrt{2}} + \lbrack 2x\rbrack_{\sqrt{2}}^{\sqrt{3}} + \lbrack 3{x\rbrack}_{\sqrt{3}}^{2}$
$= 5 - \sqrt{3} - \sqrt{2}$