Calculus III - Line Integrals - Part I (Assignment Problems)

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Section 16.2 : Line Integrals - Part I

For problems 1 – 10 evaluate the given line integral. Follow the direction of $C$ as given in the problem statement.

Evaluate $ \displaystyle \int\limits_{C}{{3y\,ds}}$ where $C$ is the portion of $x = 9 - {y^2}$ from $y = - 1$ and $y = 2$.
Evaluate $ \displaystyle \int\limits_{C}{{\sqrt x + 2xy\,ds}}$ where $C$ is the line segment from $\left( {7,3} \right)$ to $\left( {0,6} \right)$.
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.
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$.
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. 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$.
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.
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.
$The full curve starts with the graph of $y=3+x^{2}$ starting at (-2,7) and ending at (0,3). The next portion is the graph of $y=3-x^{2}$ starting at (0,3) and ending at (2,-1). The final portion is a line starting at (2,-1) and ending at (-1,-2).$
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.
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.
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 $x = 0$.
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 $\left( {6,6} \right)$.
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 $\left( {1,0,4} \right)$.