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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 16.2 : Line Integrals - Part I
For problems 1 – 10 evaluate the given line integral. Follow the direction of \(C\) as given in the problem statement.
- Evaluate \( \displaystyle \int\limits_{C}{{3y\,ds}}\) where \(C\) is the portion of \(x = 9 - {y^2}\) from \(y = - 1\) and \(y = 2\).
- Evaluate \( \displaystyle \int\limits_{C}{{\sqrt x + 2xy\,ds}}\) where \(C\) is the line segment from \(\left( {7,3} \right)\) to \(\left( {0,6} \right)\).
- 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.
- 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 \( - 1 \le t \le 0\).
- 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. 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 \( - 2 \le t \le 1\).
- 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.
- 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.
- 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.
- 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.
- Evaluate \( \displaystyle \int\limits_{C}{{{x^5}\,ds}}\) for each of the following curves.
- \(C\) 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 \(\left( {0,4} \right)\).
- \(C\) is the portion of \(y = 4 - {x^4}\) from \(x = - 1\) to \(x = 0\).
- Evaluate \( \displaystyle \int\limits_{C}{{3x - 6y\,ds}}\) for each of the following curves.
- \(C\) 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 \(\left( {6,6} \right)\).
- \(C\) is the line segment from \(\left( {6,0} \right)\) to \(\left( {6,6} \right)\).
- Evaluate \( \displaystyle \int\limits_{C}{{{y^2} - 3z + 2\,ds}}\) for each of the following curves.
- \(C\) is the line segment from \(\left( {1,0,4} \right)\) to \(\left( {2, - 1,1} \right)\).
- \(C\) is the line segment from \(\left( {2, - 1,1} \right)\) to \(\left( {1,0,4} \right)\).