Find the value of the line integral $ \int_{c} \mathbf{F} \cdot d \mathbf{r} $, where $ \mathbf{F}(x, y)=12 x y \mathbf{i}+6 x^{2} \mathbf{j} $ and $\mathbf { r } ( t ) = 8 t \mathbf { i } + 8 t ^ { 2 } \mathbf { j } , 0 \leq t \leq 1$
(Hint: If $\mathbf { F }$ is conservative, the integration may be easier on an alternate path.)
A) 3,072
B) 192
C) 6,144
D) 6,144.
E) 6,912.

Question

Quizplus · Accepted Answer

Since $ \mathbf{F} $ is conservative, the line integral is independent of path. So, we can evaluate it on the path $ \mathbf{r}(t) = 8t \mathbf{i} + 8t^2 \mathbf{j}, 0 \leq t \leq 1 $. Then,$$\begin{split} \int_C \mathbf{F} \cdot d\mathbf{r} & = \int_0^1 \mathbf{F}(\mathbf{r}(t)) \cdot \frac{d\mathbf{r}}{dt} dt \\ & = \int_0^1 (12(8t)(8t^2) + 6(8t)^2) dt \\ & = \int_0^1 (768t^3 + 384t^2) dt \\ & = \left[ 192t^4 + 128t^3 \right]_0^1 \\ & = 192 + 128 \\ & = 320. \end{split}$$

Find the Value of the Line Integral ∫cF⋅dr \int_{c} \mathbf{F} \cdot d \mathbf{r} ∫c​F⋅dr

Find the Value of the Line Integral $\int_{c} \mathbf{F} \cdot d \mathbf{r}$