Comments on What did James Stewart mean by "the line integral reduces to an ordinary single integral in this case" ?
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What did James Stewart mean by "the line integral reduces to an ordinary single integral in this case" ?
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- Please see the question in the title, in reference to the paragraph beside my two green question marks in the image below.
- How do you symbolize "the line integral reduces to an ordinary single integral in this case"? $\int^b_a f(x {\color{goldenrod}{, 0)}} \, dx = \int^b_a f(x) \, dx $?
- From $\int^b_a f(x \color{goldenrod}{, 0)} \, dx $, how exactly do you deduce $= \int^b_a f(x) \, dx$? What warrants you to drop and disregard the $\color{goldenrod}{, 0)}$?
- I disagree that $\int^b_a f(x {\color{goldenrod}{, 0)}} \, dx = \int^b_a f(x) \, dx $ for the following reasons.
- You're starting with different functions. The LHS is a BIvariate function, and the RHS is a UNIvariate function.
- The left side requires you to evaluate $f(x, y)$ at $y = 0$. $f(x)$ requires no evaluation!
I scanned James Stewart, Daniel Clegg, Saleem Watson's Calculus Early Transcendentals, 9 edn 2021, pp. 1132-3.
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