## Related questions with answers

In following problem, with $\mathbf{F}$ and $C$ as given, evaluate $\int_C \mathbf{F}(\mathbf{r}) \cdot d \mathbf{r}$ by the method that seems most suitable (direct integration, use of exactness or Green's theorem or Stokes's theorem). Recall that if $\mathbf{F}$ is a force, the integral gives the work done in a displacement. (Show the details of your work.) $\mathbf{F}=[x y, \quad z \quad 0], \quad C: y=2 x^2, \quad z=x$ from $(1,2,1)$ to $(2,8,2)$

Solution

VerifiedBy considering $x=t$ the parametric equation of the given curve $C$ can be written as

$\color{#4257b2}\mathbf{r}(t)=\left[t,~~2t^2,~~t\right]$

where $1\leq t\leq 2$.

Now by substituting $x=t$, $y=2t^2$ and $z=t$ in the given vector field, in terms of $t$ the given vector field can be written as

$\color{#4257b2}\mathbf{F}=\left[2t^3,~~t,~~0\right]$

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