Pauls Online Notes
Pauls Online Notes
Home / Calculus III / Line Integrals / Line Integrals - Part II
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.
Assignment Problems Notice
Please do not email me to get solutions and/or answers to these problems. I will not give them out under any circumstances nor will I respond to any requests to do so. The intent of these problems is for instructors to use them for assignments and having solutions/answers easily available defeats that purpose.

Section 5-3 : Line Integrals - Part II

For problems 1 – 7 evaluate the given line integral. Follow the direction of \(C\) as given in the problem statement.

  1. Evaluate \( \displaystyle \int\limits_{C}{{xy\,dx + \left( {x - y} \right)\,dy}}\) where \(C\) is the line segment from \(\left( {0, - 3} \right)\) to \(\left( { - 4,1} \right)\).
  2. Evaluate \( \displaystyle \int\limits_{C}{{{{\bf{e}}^{3x}}\,dx}}\) where \(C\) is portion of \(x = \sin \left( {4y} \right)\) from \(\displaystyle y = \frac{\pi }{8}\) to \(y = \pi \).
  3. Evaluate \( \displaystyle \int\limits_{C}{{x\,dy - \left( {{x^2} + y} \right)\,dx}}\) where \(C\) is portion of the circle centered at the origin of radius 3 in the 2nd quadrant with clockwise rotation.
  4. Evaluate \( \displaystyle \int\limits_{C}{{dx - 3{y^3}\,dy}}\) where \(C\) is given by \(\vec r\left( t \right) = 4\sin \left( {\pi t} \right)\,\,\vec i + {\left( {t - 1} \right)^2}\vec j\) with \(0 \le t \le 1\).
  5. Evaluate \( \displaystyle \int\limits_{C}{{4{y^2}\,dx + 3x\,dy + 2z\,dz}}\) where \(C\) is the line segment from \(\left( {4, - 1,2} \right)\) to \(\left( {1,7, - 1} \right)\).
  6. Evaluate \( \displaystyle \int\limits_{C}{{\left( {yz + x} \right)dx + yz\,dy\, - \left( {y + z} \right)\,dz}}\) where \(C\) is given by \(\vec r\left( t \right) = 3t\,\vec i + 4\sin \left( t \right)\vec j + 4\cos \left( t \right)\,\vec k\) with \(0 \le t \le \pi \).
  7. Evaluate \( \displaystyle \int\limits_{C}{{7xy\,dy}}\) where \(C\) is the portion of \(y = \sqrt {{x^2} + 5} \) from \(x = - 1\) to \(x = 2\) followed by the line segment from \(\left( {2,3} \right)\) to \(\left( {4, - 1} \right)\). See the sketch below for the direction.
    This curve starts with the portion of $y=\sqrt[x^{2}+5]$ starting at $\left( -1, \sqrt[6] \right)$ and ending at (2,3) and followed by a line starting at (2,3) and ending at (4,-1).
  8. Evaluate \( \displaystyle \int\limits_{C}{{\left( {{y^2} - x} \right)\,dx - 4y\,dy}}\) where \(C\) is the portion of \(y = {x^2}\) from \(x = - 2\) to \(x = 2\) followed by the line segment from \(\left( {2,4} \right)\) to \(\left( {0,6} \right)\) which in turn is followed by the line segment from \(\left( {0,6} \right)\) to \(\left( { - 2,4} \right)\). See the sketch below for the direction.
    This curve starts with the portion of $y=x^{2}$ starting at (-2,4) and ending at (2,4).  This is followed by a line starting at (2,4) and ending at (0,6).  The final portion of the curve is a line starting at (0,6) and ending at (-2,4).
  9. Evaluate \( \displaystyle \int\limits_{C}{{\left( {{x^2} - 2} \right)\,dx + 7x{y^2}\,dy}}\) for each of the following curves.
    1. \(C\) is the portion of \(x = - {y^2}\) from \(y = - 1\) to \(y = 1\).
    2. \(C\) is the line segment from \(\left( { - 1, - 1} \right)\) to \(\left( {1,1} \right)\).
  10. Evaluate \( \displaystyle \int\limits_{C}{{{x^3} + 9y\,dy}}\) for each of the following curves.
    1. \(C\) is the portion of \(y = 1 - {x^2}\) from \(x = - 1\) to \(x = 1\).
    2. \(C\) is the line segment from \(\left( { - 1,0} \right)\) to \(\left( {0, - 1} \right)\) followed by the line segment from \(\left( {0, - 1} \right)\) to \(\left( {1,0} \right)\).
  11. Evaluate \( \displaystyle \int\limits_{C}{{x{y^3}\,dx - 4x\,dy}}\) for each of the following curves.
    1. \(C\) is the portion of the circle centered at the origin of radius 7 in the 1st quadrant with counter clockwise rotation.
    2. \(C\) is the portion of the circle centered at the origin of radius 7 in the 1st quadrant with clockwise rotation.