Find the work done by F over the curve in the direction of increasing t.
-$$\mathbf { F } = x y \mathbf { i } + 3 \mathbf { j } + 8 x \mathbf { k } ; \mathrm { C } : \mathbf { r } ( \mathrm { t } ) = \cos 7 \mathrm { ti } + \sin 7 \mathrm { tj } + \mathrm { t } \mathbf { k } , 0 \leq \mathrm { t } \leq \frac { \pi } { 14 }$$
A) $W = \frac { 86 } { 21 }$
B) $W = \frac { 94 } { 21 }$
C) $W = 0$
D) $W = \frac { 80 } { 21 }$

Question

Quizplus · Accepted Answer

The work done by a force field $\mathbf{F}$ along a curve $\mathbf{r}(t)$ is given by the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$. To compute this, we first find the derivative of $\mathbf{r}(t)$ with respect to $t$, which gives us $d\mathbf{r} = (-7\sin(7t)\mathbf{i} + 7\cos(7t)\mathbf{j} + \mathbf{k})dt$. Substituting $\mathbf{r}(t)$ into $\mathbf{F}$ gives us $\mathbf{F}(t) = \cos(7t)\sin(7t)\mathbf{i} + 3\mathbf{j} + 8\cos(7t)\mathbf{k}$. The dot product $\mathbf{F} \cdot d\mathbf{r}$ then becomes $(-7\cos(7t)\sin(7t)\sin(7t) + 21\cos(7t) + 8)dt$. Integrating this expression from $0$ to $\frac{\pi}{14}$ gives the work done, which calculates to $\frac{80}{21}$.

Find the Work Done by F Over the Curve in the Direction