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### Section 5-3 : Line Integrals - Part II

For problems 1 – 5 evaluate the given line integral. Follow the direction of $$C$$ as given in the problem statement.

1. Evaluate $$\displaystyle \int\limits_{C}{{\sqrt {1 + y} \,dy}}$$ where $$C$$ is the portion of $$y = {{\bf{e}}^{2x}}$$ from $$x = 0$$ to $$x = 2$$. Solution
2. Evaluate $$\displaystyle \int\limits_{C}{{2y\,dx + \left( {1 - x} \right)\,dy}}$$ where $$C$$ is portion of $$y = 1 - {x^3}$$ from $$x = - 1$$ to $$x = 2$$. Solution
3. Evaluate $$\displaystyle \int\limits_{C}{{{x^2}\,dy - yz\,dz}}$$ where $$C$$ is the line segment from $$\left( {4, - 1,2} \right)$$ to $$\left( {1,7, - 1} \right)$$. Solution
4. Evaluate $$\displaystyle \int\limits_{C}{{1 + {x^3}\,dx}}$$ where $$C$$ is the right half of the circle of radius 2 with counter clockwise rotation followed by the line segment from $$\left( {0,2} \right)$$ to $$\left( { - 3, - 4} \right)$$. See the sketch below for the direction. Solution
5. Evaluate $$\displaystyle \int\limits_{C}{{2{x^2}\,dy - xy\,dx}}$$ where $$C$$ is the line segment from $$\left( {1, - 5} \right)$$ to $$\left( { - 2, - 3} \right)$$ followed by the portion of $$y = 1 - {x^2}$$ from $$x = - 2$$ to $$x = 2$$ which in turn is followed by the line segment from $$\left( {2, - 3} \right)$$ to $$\left( {4, - 3} \right)$$. See the sketch below for the direction. Solution
6. Evaluate $$\displaystyle \int\limits_{C}{{\left( {x - y} \right)\,dx - y{x^2}\,dy}}$$ for each of the following curves.
1. $$C$$ is the portion of the circle of radius 6 in the 1st, 2nd and 3rd quadrant with clockwise rotation.
2. $$C$$ is the line segment from $$\left( {0, - 6} \right)$$ to $$\left( {6,0} \right)$$.
Solution
7. Evaluate $$\displaystyle \int\limits_{C}{{{x^3}\,dy - \left( {y + 1} \right)\,dx}}$$ for each of the following curves.
1. $$C$$ is the line segment from $$\left( {1,7} \right)$$ to $$\left( { - 2,4} \right)$$.
2. $$C$$ is the line segment from $$\left( { - 2,4} \right)$$ to $$\left( {1,7} \right)$$.
Solution