Calculus III - Fundamental Theorem for Line Integrals (Assignment Problems)

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Section 16.5 : Fundamental Theorem for Line Integrals

Evaluate $\displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}$ where $f\left( {x,y} \right) = 5x - {y^2} + 10xy + 9$ and $C$ is given by $\displaystyle \vec r\left( t \right) = \left\langle {\frac{{2t}}{{{t^2} + 1}},1 - 8t} \right\rangle$ with $- 2 \le t \le 0$ .
Evaluate $\displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}$ where $\displaystyle f\left( {x,y,z} \right) = \frac{{3x - 8y}}{{z - 6}}$ and $C$ is given by $\vec r\left( t \right) = 6t\,\vec i + 4\vec j + \left( {9 - {t^3}} \right)\vec k$ with $- 1 \le t \le 3$ .
Evaluate $\displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}$ where $f\left( {x,y} \right) = 20y\cos \left( {x + 3} \right) - y{x^3}$ and $C$ is right half of the ellipse given by $\displaystyle {\left( {x + 3} \right)^2} + \frac{{{{\left( {y - 1} \right)}^2}}}{{16}} = 1$ with clockwise rotation.
Compute $\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ where $\vec F = 2x\,\vec i + 4y\,\vec j$ and $C$ is the circle centered at the origin of radius 5 with the counter clockwise rotation. Is $\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ independent of path? If it is not possible to determine if $\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ is independent of path clearly explain why not.
Compute $\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ where $\vec F = y\,\vec i + {x^2}\,\vec j$ and $C$ is the circle centered at the origin of radius 5 with the counter clockwise rotation. Is $\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ independent of path? If it is not possible to determine if $\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ is independent of path clearly explain why not.
Evaluate $\displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}$ where $f\left( {x,y,z} \right) = z{x^2} + x{\left( {y - 2} \right)^2}$ and $C$ is the line segment from $\left( {1,2,0} \right)$ to $\left( { - 3,10,9} \right)$ followed by the line segment from $\left( { - 3,10,9} \right)$ to $\left( {6,0,2} \right)$ .
Evaluate $\displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}$ where $f\left( {x,y} \right) = 4x + 3x{y^2} - \ln \left( {{x^2} + {y^2}} \right)$ and $C$ is the upper half of ${x^2} + {y^2} = 1$ with clockwise rotation followed by the right half of $\displaystyle {\left( {x - 1} \right)^2} + \frac{{{{\left( {y - 2} \right)}^2}}}{4} = 1$ with counter clockwise rotation. See the sketch below.
$This curve starts with the upper half of the circle of radius 1 centered at the origin with clockwise rotation followed by the portion of ${{\left( x-1 \right)}^{2}}+\frac{{{\left( y-2 \right)}^{2}}}{4}=1$ starting at (1,0) and ending at (1,4).$