Section 16.2 : Line Integrals - Part I

For problems 1 – 7 evaluate the given line integral. Follow the direction of $C$ as given in the problem statement.

1. Evaluate $\displaystyle \int\limits_{C}{{3{x^2} - 2y\,ds}}$ where $C$ is the line segment from $\left( {3,6} \right)$ to $\left( {1, - 1} \right)$. Solution
2. Evaluate $\displaystyle \int\limits_{C}{{2y{x^2} - 4x\,ds}}$ where $C$ is the lower half of the circle centered at the origin of radius 3 with clockwise rotation. Solution
3. Evaluate $\displaystyle \int\limits_{C}{{6x\,ds}}$ where $C$ is the portion of $y = {x^2}$ from $x = - 1$ to $x = 2$. The direction of $C$ is in the direction of increasing $x$. Solution
4. Evaluate $\displaystyle \int\limits_{C}{{xy - 4z\,ds}}$ where $C$ is the line segment from $\left( {1,1,0} \right)$ to $\left( {2,3, - 2} \right)$. Solution
5. Evaluate $\displaystyle \int\limits_{C}{{{x^2}{y^2}\,ds}}$ where $C$ is the circle centered at the origin of radius 2 centered on the $y$-axis at $y = 4$. See the sketches below for orientation. Note the “odd” axis orientation on the 2D circle is intentionally that way to match the 3D axis the direction.
   
   Solution
6. Evaluate $\displaystyle \int\limits_{C}{{16{y^5}\,ds}}$ where $C$ is the portion of $x = {y^4}$ from $y = 0$ to $y = 1$ followed by the line segment from $\left( {1,1} \right)$ to $\left( {1, - 2} \right)$ which in turn is followed by the line segment from $\left( {1, - 2} \right)$ to $\left( {2,0} \right)$. See the sketch below for the direction.
   
   Solution
7. Evaluate $\displaystyle \int\limits_{C}{{4y - x\,ds}}$ where $C$ is the upper portion of the circle centered at the origin of radius 3 from $\displaystyle\left( {\frac{3}{{\sqrt 2 }},\frac{3}{{\sqrt 2 }}} \right)$ to $\displaystyle\left( { - \frac{3}{{\sqrt 2 }}, - \frac{3}{{\sqrt 2 }}} \right)$ in the counter clockwise rotation followed by the line segment from $\displaystyle\left( { - \frac{3}{{\sqrt 2 }}, - \frac{3}{{\sqrt 2 }}} \right)$ to $\displaystyle\left( {4, - \frac{3}{{\sqrt 2 }}} \right)$ which in turn is followed by the line segment from $\displaystyle\left( {4, - \frac{3}{{\sqrt 2 }}} \right)$ to $\left( {4,4} \right)$. See the sketch below for the direction.
   
   Solution
8. Evaluate $\displaystyle \int\limits_{C}{{{y^3} - {x^2}\,ds}}$ for each of the following curves.
   1. $C$ is the line segment from $\left( {3,6} \right)$ to $\left( {0,0} \right)$ followed by the line segment from $\left( {0,0} \right)$ to $\left( {3, - 6} \right)$.
   2. $C$ is the line segment from $\left( {3,6} \right)$ to $\left( {3, - 6} \right)$.
   Solution
9. Evaluate $\displaystyle \int\limits_{C}{{4{x^2}\,ds}}$ for each of the following curves.
   1. $C$ is the portion of the circle centered at the origin of radius 2 in the 1^st quadrant rotating in the clockwise direction.
   2. $C$ is the line segment from $\left( {0,2} \right)$ to $\left( {2,0} \right)$.
   Solution
10. Evaluate $\displaystyle \int\limits_{C}{{2{x^3}\,ds}}$ for each of the following curves.
    1. $C$ is the portion $y = {x^3}$ from $x = - 1$ to $x = 2$.
    2. $C$ is the portion $y = {x^3}$ from $x = 2$ to $x = - 1$.
    Solution