The acceleration of a particle is $\overrightarrow{\mathrm{a}}=3{\mathrm{t}}^2\hat{\mathrm{i}}+5\mathrm{t}\hat{\mathrm{j}}-\left(8{\mathrm{t}}^3+400\right)\hat{\mathrm{k}}{\mathrm{ms}}^{-2}$. The change in velocity from t = 0 to t = 10 s is ${\mathrm{n}}_1\hat{\mathrm{i}}+{\mathrm{n}}_2\hat{\mathrm{j}}-{\mathrm{n}}_3\hat{\mathrm{k}}$ 

$\begin{array}{l}{\mathrm{dv}}_{\mathrm{x}}={\int }_0^{10} {\mathrm{a}}_{\mathrm{x}}\mathrm{dt}\\ \therefore {\mathrm{\Delta v}}_{\mathrm{x}}={\int }_0^{10} 3{\mathrm{t}}^2\mathrm{dt}={\left[{\mathrm{t}}^3\right]}_0^{10}\\ =1000{\mathrm{ms}}^{-1}\\ {\mathrm{\Delta v}}_{\mathrm{y}}={\int }_0^{10} 5\mathrm{tdt}\\ =5{\left[\frac{{\mathrm{t}}^2}{2}\right]}_0^{10}=250{\mathrm{ms}}^{-1}\\ \text{ and }{\mathrm{\Delta v}}_{\mathrm{z}}=-{\int }_0^{10} \left(8{\mathrm{t}}^3+400\right)\mathrm{dt}\\ ={\left[\frac{8{\mathrm{t}}^4}{4}+400\mathrm{t}\right]}_0^{10}\\ =-\left(2\times {10}^4+4000\right)\\ =-(20000+4000)\\ =-24000{\mathrm{ms}}^{-1}\end{array}$