Section 16.3 : 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 1^st, 2^nd and 3^rd 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