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

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Section 5-3 : Line Integrals - Part II

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

Evaluate $ \displaystyle \int\limits_{C}{{xy\,dx + \left( {x - y} \right)\,dy}}$ where $C$ is the line segment from $\left( {0, - 3} \right)$ to $\left( { - 4,1} \right)$.
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 $.
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 2^nd quadrant with clockwise rotation.
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 $0 \le t \le 1$.
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 $\left( {1,7, - 1} \right)$.
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 $.
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.
$This curve starts with the portion of $y=\sqrt[x^{2}+5]$ starting at $\left( -1, \sqrt[6] \right)$ and ending at (2,3) and followed by a line starting at (2,3) and ending at (4,-1).$
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.
$This curve starts with the portion of $y=x^{2}$ starting at (-2,4) and ending at (2,4). This is followed by a line starting at (2,4) and ending at (0,6). The final portion of the curve is a line starting at (0,6) and ending at (-2,4).$
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 $\left( {1,1} \right)$.
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 $\left( {1,0} \right)$.
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 1^st quadrant with counter clockwise rotation.
2. $C$ is the portion of the circle centered at the origin of radius 7 in the 1^st quadrant with clockwise rotation.