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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 \(\vec F\left( {x,y} \right) = {y^2}\,\vec i + \left( {3x - 6y} \right)\vec j\) and \(C\) is the line segment from \(\left( {3,7} \right)\) to \(\left( {0,12} \right)\). Solution
  2. Evaluate \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) where \(\vec F\left( {x,y} \right) = \left( {x + y} \right)\,\vec i + \left( {1 - x} \right)\vec j\) and \(C\) is the portion of \(\displaystyle\frac{{{x^2}}}{4} + \frac{{{y^2}}}{9} = 1\) that is in the 4th quadrant with the counter clockwise rotation. Solution
  3. Evaluate \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) where \(\vec F\left( {x,y} \right) = {y^2}\,\vec i + \left( {{x^2} - 4} \right)\vec j\) and \(C\) is the portion of \(y = {\left( {x - 1} \right)^2}\) from \(x = 0\) to \(x = 3\). Solution
  4. Evaluate \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) where \(\vec F\left( {x,y,z} \right) = {{\bf{e}}^{2x}}\,\vec i + z\left( {y + 1} \right)\vec j + {z^3}\,\vec k\) and \(C\) is given by \(\vec r\left( t \right) = {t^3}\,\vec i + \left( {1 - 3t} \right)\vec j + {{\bf{e}}^t}\,\vec k\) for \(0 \le t \le 2\). Solution
  5. Evaluate \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) where \(\vec F\left( {x,y} \right) = 3y\,\vec i + \left( {{x^2} - y} \right)\vec j\) and \(C\) is the upper half of the circle centered at the origin of radius 1 with counter clockwise rotation and the portion of \(y = {x^2} - 1\) from \(x = - 1\) to \(x = 1\). See the sketch below.
    Solution
  6. Evaluate \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) where \(\vec F\left( {x,y} \right) = xy\,\vec i + \left( {1 + 3y} \right)\vec j\) and \(C\) is the line segment from \(\left( {0, - 4} \right)\) to \(\left( { - 2, - 4} \right)\) followed by portion of \(y = - {x^2}\) from \(x = - 2\) to \(x = 2\) which is in turn followed by the line segment from \(\left( {2, - 4} \right)\) to \(\left( {5,1} \right)\). See the sketch below.
    Solution
  7. Evaluate \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) where \(\vec F\left( {x,y} \right) = \left( {6x - 2y} \right)\,\vec i + {x^2}\vec j\) for each of the following curves.
    1. \(C\) is the line segment from \(\left( {6, - 3} \right)\) to \(\left( {0,0} \right)\) followed by the line segment from \(\left( {0,0} \right)\) to \(\left( {6,3} \right)\).
    2. \(C\) is the line segment from \(\left( {6, - 3} \right)\) to \(\left( {6,3} \right)\).
    Solution
  8. Evaluate \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) where \(\vec F\left( {x,y} \right) = 3\,\vec i + \left( {xy - 2x} \right)\vec j\) for each of the following curves.
    1. \(C\) is the upper half of the circle centered at the origin of radius 4 with counter clockwise rotation.
    2. \(C\) is the upper half of the circle centered at the origin of radius 4 with clockwise rotation.
    Solution