VC2.4

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r\cdot s=(ti-t^{2}j+(t-1)k)\cdot (2t^{2}i+6tk)\,

=2t^{3}-t^{2}+6t(t-1)\,

\int _{0}^{2}r\cdot s\,dt=\int _{0}^{2}{2t^{3}-t^{2}+6t^{2}-6t}\,dt\,

[{\frac  {t^{4}}{2}}-{\frac  {t^{3}}{3}}+2t^{3}-3t^{2}]\, At [0,2]

8+16-12=12\,

r\times s={\begin{vmatrix}i&j&k\\t&-t^{2}&t-1\\2t^{2}&0&6t\end{vmatrix}}\,

=-6t^{3}i+[6t^{2}-2t^{2}(t-1)]j+2t^{4}k\,

=-6t^{3}i-(8t^{2}-2t^{3})j+2t^{4}k\,

Therefore,

\int _{0}^{2}r\times s\,dt=\int _{0}^{2}[-6t^{3}i-(8t^{2}-2t^{3})j+2t^{4}k]\,dt\,

=-6i\int _{0}^{2}t^{3}\,dt-j\int _{0}^{2}(8t^{2}-2t^{3})\,dt+2k\int _{0}^{2}t^{4}\,dt\,

=-6i[{\frac  {t^{4}}{4}}]-j[{\frac  {8t^{3}}{3}}-{\frac  {2t^{4}}{4}}]+2k[{\frac  {t^{5}}{5}}]\, At [0,2]

=-24i-{\frac  {40}{3}}j+{\frac  {64}{5}}k\,

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