$$\mathrm{evaluate}\:{f}\left({a},{b}\right)=\underset{{a}} {\overset{{b}} {\int}}\left({b}−{x}\right)\left({x}−{a}\right){dx} \\$$
$$\left({b}−{x}\right)\left({x}−{a}\right)=−{x}^{\mathrm{2}} +\left({a}+{b}\right){x}−{ab} \\$$$$\int_{{a}} ^{{b}} \left(−{x}^{\mathrm{2}} +\left({a}+{b}\right){x}−{ab}\right){dx} \\$$$$=\left[\frac{−{x}^{\mathrm{3}} }{\mathrm{3}}+\frac{\left({a}+{b}\right){x}^{\mathrm{2}} }{\mathrm{2}}−{abx}\right]_{{a}} ^{{b}} \\$$$$=\left[\frac{−{b}^{\mathrm{3}} }{\mathrm{3}}+\frac{\left({a}+{b}\right){b}^{\mathrm{2}} }{\mathrm{2}}−{ab}^{\mathrm{2}} \right]−\left[\frac{−{a}^{\mathrm{3}} }{\mathrm{3}}+\frac{\left({a}+{b}\right){a}^{\mathrm{2}} }{\mathrm{2}}−{a}^{\mathrm{2}} {b}\right] \\$$$$=\left[\frac{{b}^{\mathrm{3}} }{\mathrm{6}}−\frac{{ab}^{\mathrm{2}} }{\mathrm{2}}\right]−\left[\frac{{a}^{\mathrm{3}} }{\mathrm{6}}−\frac{{a}^{\mathrm{2}} {b}}{\mathrm{2}}\right] \\$$$$=\frac{{b}^{\mathrm{3}} −{a}^{\mathrm{3}} −\mathrm{3}{ab}^{\mathrm{2}} +\mathrm{3}{a}^{\mathrm{2}} {b}}{\mathrm{6}}=\frac{\left({b}−{a}\right)^{\mathrm{3}} }{\mathrm{6}} \\$$$${f}\left({a},{b}\right)=\frac{\left({b}−{a}\right)^{\mathrm{3}} }{\mathrm{6}} \\$$$$\mathrm{S}{o}\:{f}\left({a},{b}\right)\neq{f}\left({b},{a}\right) \\$$