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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.  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. 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. 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 $$\left( {1,0,4} \right)$$.