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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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  1. Please see the question in the title, in reference to the paragraph beside my two green question marks in the image below.
  2. 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 $?
  3. 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 $\color{goldenrod}{, 0)}$?
  4. I disagree that $\int^b_a f(x {\color{goldenrod}{, 0)}} \, dx = \int^b_a f(x) \, dx $ for the following reasons.
    1. You're starting with different functions. The LHS is a BIvariate function, and the RHS is a UNIvariate function.
    2. 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.

Pages 1132 and 1133 of Calculus Early Transcendentals

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3 comment threads

wrong interpretation of the phrase in 2 (1 comment)
Title equation doesn't appear in excerpt (2 comments)
Please do not use pictures for critical portions of your post. Pictures may not be legible, cannot be... (1 comment)
wrong interpretation of the phrase in 2
Snoopy‭ wrote about 1 year ago

The answer to your second bullet point is NO already. (See my answer below.) You don't need to write 3 and 4, which are all based on a wrong interpretation of the phrase in 2.