Section 16.4 : Line Integrals of Vector Fields

1. Evaluate $\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ where $\vec F\left( {x,y} \right) = {y^2}\,\vec i + \left( {3x - 6y} \right)\vec j$ and $C$ is the line segment from $\left( {3,7} \right)$ to $\left( {0,12} \right)$. Solution
2. Evaluate $\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ where $\vec F\left( {x,y} \right) = \left( {x + y} \right)\,\vec i + \left( {1 - x} \right)\vec j$ and $C$ is the portion of $\displaystyle\frac{{{x^2}}}{4} + \frac{{{y^2}}}{9} = 1$ that is in the 4^th quadrant with the counter clockwise rotation. Solution
3. Evaluate $\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ where $\vec F\left( {x,y} \right) = {y^2}\,\vec i + \left( {{x^2} - 4} \right)\vec j$ and $C$ is the portion of $y = {\left( {x - 1} \right)^2}$ from $x = 0$ to $x = 3$. Solution
4. Evaluate $\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ where $\vec F\left( {x,y,z} \right) = {{\bf{e}}^{2x}}\,\vec i + z\left( {y + 1} \right)\vec j + {z^3}\,\vec k$ and $C$ is given by $\vec r\left( t \right) = {t^3}\,\vec i + \left( {1 - 3t} \right)\vec j + {{\bf{e}}^t}\,\vec k$ for $0 \le t \le 2$. Solution
5. Evaluate $\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ where $\vec F\left( {x,y} \right) = 3y\,\vec i + \left( {{x^2} - y} \right)\vec j$ and $C$ is the upper half of the circle centered at the origin of radius 1 with counter clockwise rotation and the portion of $y = {x^2} - 1$ from $x = - 1$ to $x = 1$. See the sketch below.
   
   Solution
6. Evaluate $\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ where $\vec F\left( {x,y} \right) = xy\,\vec i + \left( {1 + 3y} \right)\vec j$ and $C$ is the line segment from $\left( {0, - 4} \right)$ to $\left( { - 2, - 4} \right)$ followed by portion of $y = - {x^2}$ from $x = - 2$ to $x = 2$ which is in turn followed by the line segment from $\left( {2, - 4} \right)$ to $\left( {5,1} \right)$. See the sketch below.
   
   Solution
7. Evaluate $\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ where $\vec F\left( {x,y} \right) = \left( {6x - 2y} \right)\,\vec i + {x^2}\vec j$ for each of the following curves.
   1. $C$ is the line segment from $\left( {6, - 3} \right)$ to $\left( {0,0} \right)$ followed by the line segment from $\left( {0,0} \right)$ to $\left( {6,3} \right)$.
   2. $C$ is the line segment from $\left( {6, - 3} \right)$ to $\left( {6,3} \right)$.
   Solution
8. Evaluate $\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ where $\vec F\left( {x,y} \right) = 3\,\vec i + \left( {xy - 2x} \right)\vec j$ for each of the following curves.
   1. $C$ is the upper half of the circle centered at the origin of radius 4 with counter clockwise rotation.
   2. $C$ is the upper half of the circle centered at the origin of radius 4 with clockwise rotation.
   Solution