Find the work done by F over the curve in the direction of increasing t.
-$$\mathbf { F } = 3 \mathrm { y } \mathbf { i } + \sqrt { \mathrm { z } } \mathbf { j } + ( 4 \mathrm { x } + 3 \mathrm { z } ) \mathbf { k } ; \mathrm { C } : \mathbf { r } ( \mathrm { t } ) = \mathrm { t } \mathbf { i } + \mathrm { t } ^ { 2 } \mathbf { j } + \mathrm { t } , 0 \leq \mathrm { t } \leq 2$$
A) $W = 22 + 20 \sqrt { 2 }$
B) $W = 31 + 20 \sqrt { 2 }$
C) $W = 22 + \frac { 16 \sqrt { 2 } } { 5 }$
D) $W = 0$

Question

Quizplus · Accepted Answer

The work done by a vector field F over a curve C is given by the line integral
$$\int_{C}\mathbf{F}\cdot d\mathbf{r}$$
where we can parameterize the curve C as $\mathbf{r}(t)=\mathrm{t}\ \mathbf{i}+{\mathrm{t}}^{2}\mathbf{j}+\mathrm{t}\ \mathbf{k}$, for $0\leq t\leq2$ and $d\mathbf{r}=\mathbf{r}'(t)dt$. Thus,
$$\int_{C}\mathbf{F}\cdot d\mathbf{r}=\int_{0}^{2}\mathbf{F}\cdot \mathbf{r}'(t)\ dt$$
$$\int_{C}\mathbf{F}\cdot d\mathbf{r}=\int_{0}^{2}(3t)(\mathbf{i}\cdot \mathbf{i})+(\sqrt{t})({\mathbf{j}}\cdot {\mathbf{j}})+(4t+3t)(\mathbf{k}\cdot \mathbf{k})\ dt$$
$$\int_{C}\mathbf{F}\cdot d\mathbf{r}=\int_{0}^{2}(3t+3t+7t^{3/2})\ dt$$
$$\int_{C}\mathbf{F}\cdot d\mathbf{r}= \left[ \frac { 7 } { 5 } t^ { 5 / 2 } + \frac { 3 } { 2 } t ^2  \right]_{0}^{2}= 22+\frac{16\sqrt{2}}{5}$$

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