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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 $$\vec F\left( {x,y} \right) = 2{x^2}\,\vec i + \left( {{y^2} - 1} \right)\vec j$$ and $$C$$ is the portion of $$\displaystyle \frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{9} = 1$$ that is in the 1st, 4th and 3rd quadrant with the clockwise orientation.
2. Evaluate $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where $$\vec F\left( {x,y} \right) = xy\,\vec i + \left( {4x - 2y} \right)\vec j$$ and $$C$$ is the line segment from $$\left( {4, - 3} \right)$$ to $$\left( {7,0} \right)$$.
3. Evaluate $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where $$\vec F\left( {x,y} \right) = \left( {{x^3} - y} \right)\,\vec i + \left( {{x^2} + 7x} \right)\vec j$$ and $$C$$ is the portion of $$y = {x^3} + 2$$ from $$x = - 1$$ to $$x = 2$$.
4. Evaluate $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where $$\vec F\left( {x,y} \right) = xy\,\vec i + \left( {1 + {x^2}} \right)\vec j$$ and $$C$$ is given by $$\vec r\left( t \right) = {{\bf{e}}^{6t}}\,\vec i + \left( {4 - {{\bf{e}}^{2t}}} \right)\vec j$$ for $$- 2 \le t \le 0$$.
5. Evaluate $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where $$\vec F\left( {x,y,z} \right) = \left( {3x - 3y} \right)\,\vec i + \left( {{y^3} - 10} \right)\vec j + y\,z\,\vec k$$ and $$C$$ is the line segment from $$\left( {1,4, - 2} \right)$$ to $$\left( {3,4,6} \right)$$.
6. Evaluate $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where $$\vec F\left( {x,y,z} \right) = \left( {x + z} \right)\,\vec i + {y^3}\vec j + \left( {1 - x} \right)\,\vec k$$ and $$C$$ is the portion of the spiral on the $$y$$-axis given by $$\vec r\left( t \right) = \cos \left( {2t} \right)\,\vec i - t\,\vec j + \sin \left( {2t} \right)\,\vec k$$ for $$- \pi \le t \le 2\pi$$.
7. Evaluate $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where $$\vec F\left( {x,y} \right) = {x^2}\,\vec i + \left( {{y^2} - x} \right)\vec j$$ and $$C$$ is the line segment from $$\left( {2,4} \right)$$ to $$\left( {0,4} \right)$$ followed by the line segment form $$\left( {0,4} \right)$$ to $$\left( {3, - 1} \right)$$.
8. Evaluate $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where $$\vec F\left( {x,y} \right) = xy\,\vec i - 3\,\vec j$$ and $$C$$ is the portion of $$\displaystyle {x^2} + \frac{{{y^2}}}{16} = 1$$ in the 2nd quadrant with clockwise rotation followed by the line segment from $$\left( {0,4} \right)$$ to $$\left( {4, - 2} \right)$$. See the sketch below. 9. Evaluate $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where $$\vec F\left( {x,y} \right) = x{y^2}\,\vec i + \left( {2y + 3x} \right)\vec j$$ and $$C$$ is the portion of $$x = {y^2} - 1$$ from $$y = - 2$$ to $$y = 2$$ followed by the line segment from $$\left( {3,2} \right)$$ to $$\left( {0,0} \right)$$ which in turn is followed by the line segment from $$\left( {0,0} \right)$$ to $$\left( {3, - 2} \right)$$. See the sketch below. 10. Evaluate $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where $$\vec F\left( {x,y} \right) = \left( {1 - {y^2}} \right)\,\vec i - x\,\vec j$$ for each of the following curves.
1. $$C$$ is the top half of the circle centered at the origin of radius 1 with the counter clockwise rotation.
2. $$C$$ is the bottom half of $$\displaystyle {x^2} + \frac{{{y^2}}}{{36}}$$ with clockwise rotation.
11. Evaluate $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where $$\vec F\left( {x,y} \right) = \left( {{x^2} + y + 2} \right)\,\vec i + x\,y\,\vec j$$ for each of the following curves.
1. $$C$$ is the portion of $$y = {x^2} - 2$$ from $$x = - 3$$ to $$x = 3$$.
2. $$C$$ is the line segment from $$\left( { - 3,5} \right)$$ to $$\left( {3,5} \right)$$.
12. Evaluate $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where $$\vec F\left( {x,y} \right) = {y^2}\,\vec i + \left( {1 - 3x} \right)\,\vec j$$ for each of the following curves.
1. $$C$$ is the line segment from $$\left( {1,4} \right)$$ to $$\left( { - 2,3} \right)$$.
2. $$C$$ is the line segment from $$\left( { - 2,3} \right)$$ to $$\left( {1,4} \right)$$.
13. Evaluate $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where $$\vec F\left( {x,y} \right) = - 2x\,\vec i + \left( {x + 2y} \right)\vec j$$ for each of the following curves.
1. $$C$$ is the portion of $$\displaystyle \frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{4} = 1$$ in the 1st quadrant with counter clockwise rotation.
2. $$C$$ is the portion of $$\displaystyle \frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{4} = 1$$ in the 1st quadrant with clockwise rotation.