$t \mathbf{i}\+\mathbf{j}\+\mathbf{k}$ 

which means that this particular curve would pass through $\left(0\,1\,1\right)$ at $t\=0$.

 $r\=\left[\begin{array}{c}t\\ t^k\\ t^{2 k}\end{array}\right]$ 

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Loading Student:-VectorCalculus

Define the vector field F

$⟨2 x z\+y^2\,2 x y-z^3\,x^2-3 y z^2⟩$ = $\left[\begin{array}{c}2xz+y^2\\ -z^3+2xy\\ -3y{z}^2+x^2\end{array}\right]$$\stackrel{\text{to Vector Field}}{\to }$$\left[\begin{array}{c}2xz+y^2\\ -z^3+2xy\\ -3y{z}^2+x^2\end{array}\right]$$\stackrel{\text{assign to a name}}{\to }$$F$$$

Form and evaluate the line integral of F along curves of the family

$\mathbf{F}$

$\stackrel{\text{line integral}}{\to }$

${\∫}_0^1\left(2t{t}^{2k}\+{\left(t^k\right)}^2\+\left(-{\left(t^{2k}\right)}^3\+2t{t}^k\right)t^{k-1}k\+2\left(-3t^k{\left(t^{2k}\right)}^2\+t^2\right)t^{2k-1}k\right)\ⅆt$

$\stackrel{\text{simplify}}{\=}$

${\∫}_0^1\left(-7k{t}^{7k-1}\+2k{t}^{2k\+1}\+2t^{2k}k\+2t^{2k\+1}\+t^{2k}\right)\ⅆt$

$\=$

$\underset{t\→0^{\+}}{lim}\left(t^{7k}-t^{2k\+2}-t^{2k\+1}\right)\+1$

$\stackrel{\text{assuming real range}}{\to }$

$1$

Table 9.7.6(a)  Solution via the LineInt command accessed from the Context Panel

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$\mathrm{with}\left(\mathrm{Student}:-\mathrm{VectorCalculus}\right)\:$

$\mathrm{BasisFormat}\left(\mathrm{false}\right)\:$

$\mathbf{F}≔\mathrm{VectorField}\left(⟨2 x z\+y^2\,2 x y-z^3\,x^2-3 y z^2⟩\right)\:$

Form and evaluate the line integral of F along curves of the family

$\mathrm{simplify}\left(\mathrm{LineInt}\left(\mathbf{F}\,\mathrm{Path}\left(⟨t\,t^k\,t^{2 k}⟩\,t\=0..1\right)\,\mathrm{output}\=\mathrm{integral}\right)\right)$

${\∫}_0^1\left(-7k{t}^{7k-1}\+2k{t}^{2k\+1}\+2t^{2k}k\+2t^{2k\+1}\+t^{2k}\right)\ⅆt$

$\mathrm{LineInt}\left(\mathbf{F}\,\mathrm{Path}\left(⟨t\,t^k\,t^{2 k}⟩\,t\=0..1\right)\right) assuming k\ge 1$ = $1$$$

Table 9.7.6(b)   Solution via the LineInt command