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

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.

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
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
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
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
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.
$A three dimensional sketch of the curve $C$. It is a circle of radius 2 that lies in a plane perpendicular to the y-axis and is centered on the y-axis at $y = 4$. Arrows on the circle show the direction of travel to be from the positive z-axis moving towards the positive x-axis.$
Solution
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.
$The curve $C$. The arrows show that it starts at the origin and follows the curve $x = {y^4}$ up to $\left( {1,1} \right)$, then drops straight down the line $x = 1$ to $\left( {1, - 2} \right)$ and finally runs up and to the right along a straight line to $\left( {2,0} \right)$.$
Solution
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.
$The curve $C$. The arrows show that it starts at $\left( {\frac{3}{{\sqrt 2 }},\frac{3}{{\sqrt 2 }}} \right)$ and travels counter clockwise around the circle of radius 3 centered at the origin, up over the top and around the left side, to $\left( { - \frac{3}{{\sqrt 2 }}, - \frac{3}{{\sqrt 2 }}} \right)$. From there it runs right along a horizontal line to $\left( {4, - \frac{3}{{\sqrt 2 }}} \right)$ and then straight up to $\left( {4,4} \right)$.$
Solution
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
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
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