Find the value of the line integral $\int_{c} \mathbf { F } \text {. } d \mathbf { r } \text {, where } \mathbf { F } ( x , y ) = e ^ { y } ( z \mathbf { i } + x z \mathbf { j } + x \mathbf { k } ) \text { and }$ $\mathbf { r } ( t ) = 3 \cos t \mathbf { i } + 3 \sin t \mathbf { j } + 8 \mathbf { k } , 0 \leq t \leq \pi$
(Hint: If $\mathbf { F }$ is conservative, the integration may be easier on an alternate path.)
A) - 24
B) 48
C) - 48
D) 24
E) 0

Question

Quizplus · Accepted Answer

The vector field $\mathbf{F}$ is conservative because it can be expressed as the gradient of a scalar potential function. The path integral of a conservative vector field over a closed path or between two points is zero if the start and end points are the same, or it depends only on the values of the potential function at the endpoints. Since the path is closed (a semicircle returning to the same height), the integral is zero.

Find the Value of the Line Integral r(t)=3cos⁡ti+3sin⁡tj+8k,0≤t≤π\mathbf { r } ( t ) = 3 \cos t \mathbf { i } + 3 \sin t \mathbf { j } + 8 \mathbf { k } , 0 \leq t \leq \pir(t)=3costi+3sintj+8k,0≤t≤π

Find the Value of the Line Integral $\mathbf { r } ( t ) = 3 \cos t \mathbf { i } + 3 \sin t \mathbf { j } + 8 \mathbf { k } , 0 \leq t \leq \pi$