Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where F $( x , y , z ) = y z \mathbf { i } + x z \mathbf { j } + x y \mathbf { k }$ , and the curve $C$ is given by the vector function $\mathbf { r } ( t ) = t ^ { 2 } \mathbf { i } + t \mathbf { j } + t ^ { 3 } \mathbf { k } , 0 \leq t \leq 1$ .
A) $- 3$
B)$- 2$
C)$- 1$
D)$0$
E)$1$
F)$2$
G)$3$
H) $4$

Question

Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where F $( x , y , z ) = y z \mathbf { i } + x z \mathbf { j } + x y \mathbf { k }$ , and the curve $C$ is given by the vector function $\mathbf { r } ( t ) = t ^ { 2 } \mathbf { i } + t \mathbf { j } + t ^ { 3 } \mathbf { k } , 0 \leq t \leq 1$ .
A) $- 3$ 
B)$- 2$
C)$- 1$
D)$0$
E)$1$
F)$2$ 
G)$3$
H) $4$

Quizplus · Accepted Answer

To evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$, we first substitute the parametric equations of the curve $C$ into the vector field $\mathbf { F }$. Given $\mathbf { F } ( x , y , z ) = y z \mathbf { i } + x z \mathbf { j } + x y \mathbf { k }$ and $\mathbf { r } ( t ) = t ^ { 2 } \mathbf { i } + t \mathbf { j } + t ^ { 3 } \mathbf { k }$, we have $x = t^2$, $y = t$, and $z = t^3$. Substituting these into $\mathbf { F }$, we get $\mathbf { F }(t) = t^4 \mathbf { i } + t^5 \mathbf { j } + t^3 \mathbf { k }$.Next, we find $d\mathbf { r }$ by differentiating $\mathbf { r }(t)$ with respect to $t$, which gives $d\mathbf { r } = (2t \mathbf { i } + \mathbf { j } + 3t^2 \mathbf { k })dt$.The dot product $\mathbf { F } \cdot d\mathbf { r }$ is then $(t^4)(2t) + (t^5)(1) + (t^3)(3t^2) = 2t^5 + t^5 + 3t^5 = 6t^5$.Finally, we integrate this expression from $0$ to $1$: $\int_0^1 6t^5 dt = [t^6]_0^1 = 1^6 - 0^6 = 1$.Therefore, the value of the line integral is $1$.

Evaluate the Line Integral ∫CF⋅dr\int _ { C } \mathbf { F } \cdot d \mathbf { r }∫C​F⋅dr

Evaluate the Line Integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$