Calculus III - Line Integrals of Vector Fields (Assignment Problems)

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Section 16.4 : Line Integrals of Vector Fields

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 1^st, 4^th and 3^rd quadrant with the clockwise orientation.
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 $\left( {7,0} \right)$.
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 $x = 2$.
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$.
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 $\left( {3,4,6} \right)$.
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 $.
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 $\left( {3, - 1} \right)$.
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 2^nd quadrant with clockwise rotation followed by the line segment from $\left( {0,4} \right)$ to $\left( {4, - 2} \right)$. See the sketch below.
$This curve starts with the portion of ${{x}^{2}}+\frac{{{y}^{2}}}{4}=1$ starting at (-1,0) and ending at (0,4) followed by a line starting at (0,4) and ending at (4,-2).$
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.
$This curve starts with the portion of $x=y^{2}-1$ starting at (3,-2) and ending at (3,2). This is followed by a line starting at (3,2) and ending at the origin. The final portion of the curve is a line starting at the origin and ending at (3.-2).$
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 $\displaystyle {x^2} + \frac{{{y^2}}}{{36}}=1$ with clockwise rotation.
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 $\left( {3,5} \right)$.
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 $\left( {1,4} \right)$.
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 1^st quadrant with counter clockwise rotation.
2. $C$ is the portion of $\displaystyle \frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{4} = 1$ in the 1^st quadrant with clockwise rotation.