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### Section 16.3 : 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. 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. 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.