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Section 5-2 : Line Integrals - Part I

For problems 1 – 10 evaluate the given line integral. Follow the direction of \(C\) as given in the problem statement.

  1. Evaluate \( \displaystyle \int\limits_{C}{{3y\,ds}}\) where \(C\) is the portion of \(x = 9 - {y^2}\) from \(y = - 1\) and \(y = 2\).
  2. Evaluate \( \displaystyle \int\limits_{C}{{\sqrt x + 2xy\,ds}}\) where \(C\) is the line segment from \(\left( {7,3} \right)\) to \(\left( {0,6} \right)\).
  3. Evaluate \( \displaystyle \int\limits_{C}{{{y^2} - 10xy\,ds}}\) where \(C\) is the left half of the circle centered at the origin of radius 6 with counter clockwise rotation.
  4. Evaluate \( \displaystyle \int\limits_{C}{{{x^2} - 2y\,ds}}\) where \(C\) is given by \(\vec r\left( t \right) = \left\langle {4{t^4},{t^4}} \right\rangle \) for \( - 1 \le t \le 0\).
  5. Evaluate \( \displaystyle \int\limits_{C}{{{z^3} - 4x + 2y\,ds}}\) where \(C\) is the line segment from \(\left( {2,4, - 1} \right)\) to \(\left( {1, - 1,0} \right)\).
  6. 6. Evaluate \( \displaystyle \int\limits_{C}{{x + 12xz\,ds}}\) where \(C\) is given by \(\displaystyle \vec r\left( t \right) = \left\langle {t,\frac{1}{2}{t^2},\frac{1}{4}{t^4}} \right\rangle \) for \( - 2 \le t \le 1\).
  7. Evaluate \( \displaystyle \int\limits_{C}{{{z^3}\left( {x + 7} \right) - 2y\,ds}}\) where \(C\) is the circle centered at the origin of radius 1 centered on the \(x\)-axis at \(x = - 3\) . See the sketches below for the direction.
    This is a sketch with the standard 3D coordinate system.  The positive z-axis is straight up, the positive x-axis moves off to the left and slightly downward and positive y-axis moves off the right and slightly downward.  The circle centered on the x-axis at x=-3.  If you are standing on the positive x-axis and looking towards the circle there is a clockwise orientation to the circle. This is a 2D sketch of the circle.  In this case think of the x-axis as coming directly out of the page and is not shown.  The positive z-axis is pointed upwards from the origin and the positive y-axis is pointed right out of the origin.  The circle has a clockwise orientation.
  8. Evaluate \( \displaystyle \int\limits_{C}{{6x\,ds}}\) where \(C\) is the portion of \(y = 3 + {x^2}\) from \(x = - 2\) to \(x = 0\) followed by the portion of \(y = 3 - {x^2}\) form \(x = 0\) to \(x = 2\) which in turn is followed by the line segment from \(\left( {2, - 1} \right)\) to \(\left( { - 1, - 2} \right)\). See the sketch below for the direction.
    The full curve starts with the graph of $y=3+x^{2}$ starting at (-2,7) and ending at (0,3).  The next portion is the graph of $y=3-x^{2}$ starting at (0,3) and ending at (2,-1).  The final portion is a line starting at (2,-1) and ending at (-1,-2).
  9. Evaluate \( \displaystyle \int\limits_{C}{{2 - xy\,ds}}\) where \(C\) is the upper half of the circle centered at the origin of radius 1 with the clockwise rotation followed by the line segment form \(\left( {1,0} \right)\) to \(\left( {3,0} \right)\) which in turn is followed by the lower half of the circle centered at the origin of radius 3 with the clockwise rotation. See the sketch below for the direction.
    The full curve starts with upper part of the circle centered at radius 1 centered at the origin with clockwise orientation.  This is followed by a line from (1,0) to (3,0).  Finally, there is the lower half of a circle centered at radius 3 centered at the origin with counter clockwise orientation.
  10. Evaluate \( \displaystyle \int\limits_{C}{{3xy + {{\left( {x - 1} \right)}^2}\,ds}}\) where \(C\) is the triangle with vertices \(\left( {0,3} \right)\), \(\left( {6,0} \right)\) and \(\left( {0,0} \right)\) with the clockwise rotation.
  11. Evaluate \( \displaystyle \int\limits_{C}{{{x^5}\,ds}}\) for each of the following curves.
    1. \(C\) is the line segment from \(\left( { - 1,3} \right)\) to \(\left( {0,0} \right)\) followed by the line segment from \(\left( {0,0} \right)\) to \(\left( {0,4} \right)\).
    2. \(C\) is the portion of \(y = 4 - {x^4}\) from \(x = - 1\) to \(x = 0\).
  12. Evaluate \( \displaystyle \int\limits_{C}{{3x - 6y\,ds}}\) for each of the following curves.
    1. \(C\) is the line segment from \(\left( {6,0} \right)\) to \(\left( {0,3} \right)\) followed by the line segment from \(\left( {0,3} \right)\) to \(\left( {6,6} \right)\).
    2. \(C\) is the line segment from \(\left( {6,0} \right)\) to \(\left( {6,6} \right)\).
  13. Evaluate \( \displaystyle \int\limits_{C}{{{y^2} - 3z + 2\,ds}}\) for each of the following curves.
    1. \(C\) is the line segment from \(\left( {1,0,4} \right)\) to \(\left( {2, - 1,1} \right)\).
    2. \(C\) is the line segment from \(\left( {2, - 1,1} \right)\) to \(\left( {1,0,4} \right)\).