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### Section 5-5 : Fundamental Theorem for Line Integrals

1. 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 $$- 2 \le t \le 0$$.
2. 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 $$- 1 \le t \le 3$$.
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.
4. 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.
5. 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.
6. 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 $$\left( {6,0,2} \right)$$.
7. 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.