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### 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 1st 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