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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 \(\displaystyle {x^2} + \frac{{{y^2}}}{4} = 1\) 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.
    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).
  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.
    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).
  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.