11.3: Line Integrals

1. Evaluate $\displaystyle \int_C xy^2\ ds$, where $C$ is the line segment from $(1, 2, 0)$ to $(2, 1, 3)$.
    
2. Evaluate $\displaystyle \int_C (x + y)\ ds$, where $C$ is the line segment from $(0, 1, 0)$ to $(1, 0, 0)$.
    
3. Evaluate $\displaystyle \int_C x^2yz\ ds$, where $C$ is the curve $\mathbf{r}(t) = (6t^3, 3t^2, t)$, $0 \leq t \leq 1$.
    
4. Evaluate $\displaystyle \int_C \dfrac{y}{1 + x^2}\ ds$, where $C$ is the curve $\mathbf{r}(t) = t\mathbf{i} + (1 + 2t)\mathbf{j}$, $0 \leq t \leq 1$.
    
5. Evaluate $\displaystyle \int_C \mathbf{F}\cdot d\mathbf{r}$, where $\mathbf{F}(x, y) = (\sin x, \cos y)$, and $C$ is the top half of the unit circle counterclockwise from $(1, 0)$ to $(-1, 0)$.
    
6. Evaluate $\displaystyle \int_C \mathbf{F}\cdot d\mathbf{r}$, where $\mathbf{F}(x, y) = (\sin x, \cos y)$, and $C$ is the top half of the unit circle clockwise from $(-1, 0)$ to $(1, 0)$.
    
7. Evaluate $\displaystyle \int_C (xe^y, x^2y) \cdot d\mathbf{r}$, where $C$ is the curve $\mathbf{r}(t) = (3t, t^2)$, $0 \leq t \leq 1$.
    
8. Evaluate $\displaystyle \int_C \dfrac{1}{xy}\ dx + \dfrac{1}{x + y}\ dy$, where $C$ is the path from $(1, 1)$ to $(3, 1)$ to $(3, 6)$.
    
9. Evaluate $\displaystyle \int_C x^2yz\ dx$, where $C$ is the curve $\mathbf{r}(t) = (6t^3, 3t^2, t)$, $0 \leq t \leq 1$.
    
10. Find the work done by the force field $\mathbf{F}(x, y, z) = x\mathbf{i} + 3xy\mathbf{j} - (x + z)\mathbf{k}$ on a particle moving along a line segment that goes from $(1, 4, 2)$ to $(0, 5, 1)$.
     
11. Find the work done by the force field $\mathbf{F}(x, y, z) = (y, z, x)$ on an object that moves along the curve $\mathbf{r}(t) = \left(\sqrt{t}, \dfrac{1}{\sqrt{t}}, t\right)$, $1 \leq t \leq 4$.
     
12. Compute the circulation of $\mathbf{F}(x, y) = (-2, y)$ along the circle of radius $2$ centered at the origin, oriented counterclockwise.
     
13. Compute the flux of $\mathbf{F}(x, y) = x^2\mathbf{i} + y\mathbf{j}$ across a line segment from $(0, 0)$ to $(1, 2)$.
     
14. Compute the flux of $\mathbf{F}(x, y) = (-y, x)$ across the unit circle centered at the origin, oriented counterclockwise.