QuestionEvaluate $\int F \cdot d r$ for $\bar{F}=x y \bar{i}-z j+x^{2} \bar{k}$ along the curve $x=t^{2}$, $y=2t$, $z=t^{3}$ from $t=0$ to $t=1$. Answer: $\frac{51}{70}$.

Studdy Solution
Now, evaluate the integral.
$$\int_{0}^{} (4t^{4} -2t^{3} +3t^{6}) dt = \left[\frac{4}{5}t^{5} - \frac{}{2}t^{4} + \frac{3}{7}t^{7}\right]_{0}^{} = \frac{4}{5} - \frac{}{2} + \frac{3}{7} = \frac{51}{70}$$The value of the line integral of the vector field $\bar{}$ along the curve $c$ is $\frac{51}{70}$.