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### Section 5-4 : 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 4th 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