Activity 13.5.2.
In this activity, we will be making sense of scalar line integrals by examining a few common functions and justifying whether the scalar line integrals given are positive, negative, or zero. Let the functions \(f_1\text{,}\) \(f_2\text{,}\) \(f_3\text{,}\) and \(f_4\) be defined as
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\(\displaystyle f_1(x,y,z)=y\)
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\(\displaystyle f_2(x,y,z)=z\)
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\(\displaystyle f_3(x,y,z)=x^2\)
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\(\displaystyle f_4(x,y,z)=x-y\)
(a)
For each of the paths given below, sketch (in either 2D or 3D) the curve and label at least three points on the curve including the end points (if they exist).
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\(C_1\) is the part of the unit circle in the \(xy\)-plane centered at the origin that is above the line \(y=-x\text{.}\)
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\(C_2\) is the part of the curve at the intersection of the cylinder given by \(x^2+y^2=1\) and the plane \(z=x\) such that \(y \geq -x\text{;}\) You may want to consider the circle that is the intersection of \(x^2+y^2=1\) and \(z=x\text{,}\) then think about which half of this circle satisfies the inequality \(y \geq -x\)
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\(C_3\) is the part of the helix given by \(\vr(t)=\langle \cos(t),\sin(t),\frac{t}{2 \pi}\rangle\) with \(t \in [0,\pi]\)
(b)
For each of the functions \(f_1\text{,}\) \(f_2\text{,}\) and \(f_3\) defined above, state whether \(\int_{C_1} f_i \, ds \) is positive, negative, or zero. Be sure to justify your answer in terms of the function being integrated and the particulars of the curve of integration.
(c)
For each of the functions \(f_1\text{,}\) \(f_2\text{,}\) and \(f_3\text{,}\) defined above, state whether \(\int_{C_2} f_i \, ds \) is positive, negative, or zero. Be sure to justify your answer in terms of the function being integrated and the particulars of the curve of integration.
(d)
For each of the functions \(f_1\text{,}\) \(f_2\text{,}\) and \(f_3\text{,}\) defined above, state whether \(\int_{C_3} f_i \, ds \) is positive, negative, or zero. Be sure to justify your answer in terms of the function being integrated and the particulars of the curve of integration.
(e)
For the function \(f_4\text{,}\) defined above, state each of the following integrals is positive, negative, or zero. Be sure to justify your answer in terms of the function being integrated and the particulars of the curve of integration. You should consider which parts of the curve being integrated will have positive/negative/zero output for the function \(f_4\text{.}\)
\begin{gather*}
\int_{C_1} f_4 \, ds \\
\int_{C_2} f_4 \, ds \\
\int_{C_3} f_4 \, ds \text{,}
\end{gather*}
