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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.
    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).