VC2.8

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\int _{1}^{2}r\times {\frac  {d^{2}r}{dt^{2}}}\,dt=[r\times {\frac  {dr}{dt}}]\, at [1,2]

Given,r(t)=5t^{2}i+tj-r^{3}k\,

{\frac  {dr}{dt}}=10ti+j-3t^{2}k\,

Therefore, r\times {\frac  {dr}{dt}}=(5t^{2}i+tj-r^{3}k)\times (10ti+j-3t^{2}k)\,

={\begin{vmatrix}i&j&k\\5t^{2}&t&-t^{3}\\10t&1&-3t^{2}\end{vmatrix}}\,

=-2t^{3}i+5t^{4}j-5t^{2}k\,

Therefore,\int _{1}^{2}r\times {\frac  {d^{2}r}{dt^{2}}}\,dt=[-2t^{3}i+5t^{4}j-5t^{2}k]\, at t=[1,2]

=-2(8-1)i+5(16-1)j-5(4-1)k=-14i+75j-15k\,

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