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Contents Index Calc Dark Mode Prev Up Next \(\newcommand{\R}{\mathbb{R}}
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Activity 13.5.3 .
In this activity, we will examine why we must be careful when using symmetry to make arguments about scalar line integral. Let
\(C_1\) and
\(C_2\) be the paths shown in
Figure 13.5.16 . We will consider the function
\(f(x,y)=x\) for this activity.
Figure 13.5.16. A plot of paths \(C_1\) and \(C_2\)
(a) Parameterize
\(C_1\) and
\(C_2\) as
\(\vr_1(t)\) and
\(\vr_2(t)\text{.}\) (It is fine to have
\(0\leq t\leq 1\) for both of your parameterizations.)
(b) Use
Theorem 13.5.11 to compute
\(\displaystyle\int_{C_1} f\, ds\) and
\(\displaystyle\int_{C_2} f\, ds\text{.}\)
(c) As with line integrals of vector fields, we have that
\(\displaystyle\int_{C_1+C_2} f\, ds = \int_{C_1} f\, ds + \int_{C_2} f\, ds\text{.}\) Use this property to compute
\(\displaystyle\int_{C_1+C_2} f\, ds\text{.}\)
