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In page 285 of "Differential and Integral Calculus, 6th ed." by Love and Rainville, the example used in 'Substitution suggested by the problem' about plane areas is this:

Find the area of the ellipse $(x = a \cos{\varphi}$, $y = b \sin{\varphi})$.

Then, the solution is given by substitution.

At once, \begin{gather*}A = 4\int_{0}^{a} y\,dx\end{gather*}. When $x = 0$, $\varphi = \frac{1}{2}\pi$, and when $x = a$, $\varphi = 0$. Therefore \begin{align*}A &\;=\; 4\int_{\frac{1}{2}\pi}^{0}(b\sin\varphi)(-a\sin\varphi\,d\varphi) \\ &\;=\;4ab\int^{\frac{1}{2}\pi}_{0}\sin^{2}\varphi\,d\varphi \\ &\;=\; 4ab \cdot \frac{1}{2}\cdot \frac{\pi}{2} \\ &\;=\; \pi ab.\end{align*}

It is then followed by these sentences:

The transformations suggested in this section are intuitively reasonable. Rigorous justification of them belongs to a course in advanced calculus.

Because of this, I had two questions in mind.

1. Is the "them" the given solution? [Answered by @user170231]

  1. Why does it need a rigorous justification? Isn't the given solution enough?
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  • $\begingroup$ "Them" refers to the transformations or change of variables $\endgroup$
    – user170231
    Commented Mar 3, 2021 at 15:02
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    $\begingroup$ Any definition of the "area" of a complex figure presupposes some formalities. I agree that this kind of thing is fairly close to the surface because of how scaling along axes affects the areas of rectangles. But there are regions in $\mathbb{R}^n$ where the limits you'd want to use to define the area do not exist. $\endgroup$ Commented Mar 3, 2021 at 15:08
  • $\begingroup$ I don't think it refers to complex figures. It just used the ellipse as an example to find the area using its parametric form, $\endgroup$
    – soupless
    Commented Mar 3, 2021 at 15:11
  • $\begingroup$ When I say "complex" I only mean "not a union of rectangles." Euclid had an interesting approach to area. He did not compute areas with numbers but provided constructions for splitting a figure up into pieces and reassembling the pieces into another figure. This implicit notion of area ("equidecomposability") is not quite the same as our notion. See e.g. en.wikipedia.org/wiki/Dehn_invariant $\endgroup$ Commented Mar 3, 2021 at 15:17
  • $\begingroup$ Sorry, I don't quite understand the concept of tiling, including your stated link, Dehn invariant. Is there like another reason for this? $\endgroup$
    – soupless
    Commented Mar 3, 2021 at 15:39

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"Rigorous justifications" likely means nothing given that it has it doesn't seem to have a justified context within the content of the book, unless they have left a footnote on the context they wish to express; either this, or they are leaving out elliptic integrals which may be in a further chapter and which are not necessary for the computation of the area of an ellipse. That said, I can't tell you any more about their statement beyond reasonable speculation. As observed, if

$$\left(\dfrac{x}{a}\right)^2+\left(\dfrac{y}{b}\right)^2=1,$$ then $$(x,y)=(a\cos\theta,b\sin\theta)$$ produces the correct identity $$\cos^2\theta+\sin^2\theta=1,$$ and so the transformation is justified. Moreover, it should be noted that upon substituting $$(x,y)=(a\cos\theta,b\sin\theta)$$ that the parametrization is still circular with respect to the angle, and so the bounds of integration with respect to the angle $\theta$ ranging from $0$ to $\dfrac{\pi}{2}$ produce a quarter of the area of the ellipse. The boundaries of integration can also be derived from the inverse sine function in the sense that

$$[\arcsin(0),\arcsin(1)]=[0,\dfrac{\pi}{2}]$$

when the inverse sine is restricted to the region on the plane where $x\in [-1,1], y\in [-\frac{\pi}{2},\frac{\pi}{2}]$.

It seems that the wording of the sentences "The transformations suggested in this section are intuitively reasonable. Rigorous justification of them belongs to a course in advanced calculus." are not faithful to the material they present, on the basis that there is not much further rigor to discuss here. If there is a greater context they would share in some other book or by some other mathematicians, then let them state it as such. But, as it stands (and according to their wording), math is not objective in the sense they imply; any further rigor comes from time-elaborated research-level discoveries around the context of the subject which can then be used to further re-enforce the truth of the result. The only requirements for contributing to the body of mathematics is that your discovery is new, interesting, and true by appropriately rigorous proof. For example, the logical proof that $1+1=2$ is very long, but inappropriate for an elementary school student, because the proof itself is actually the interesting result.

That said, if they have some other context in mind, then I would gladly like to learn it. Nevertheless, I am upset they decided to make such a remark. Mainly, however, I hope to reinforce your confidence in the subject and to not rely on other's remarks as considerations simply because it seems as if they demand it.

I hope this answer itself is appropriate to the question.

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  • $\begingroup$ So it is the $\left(\frac{x}{a}\right)^{2} + \left(\frac{y}{b}\right)^{2} = 1$ and the $\cos^{2}{\theta} + \sin^{2}{\theta} = 1$ that justifies the solution given rigorously. Did I understand it correctly? $\endgroup$
    – soupless
    Commented Mar 8, 2021 at 14:09
  • $\begingroup$ Yes, soupless. In my opinion that is all that is needed. $\endgroup$ Commented Mar 8, 2021 at 15:04
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    $\begingroup$ @AlexanderConrad I'm here! Fully in agreement of the second last sentence, and I don't actually think there's other context that they are thinking of. I too think that particular phrase was unnecessary. In fact the whole book lacks a bit of rigor, probably to provide a gentler introduction for engineers and other scientists. Usually such details are covered in appendices as required, not here though. +1, and in no way do I find your post opinionated enough to cause controversy, although there may be differing viewpoints on that phrase being necessary. $\endgroup$ Commented Mar 10, 2021 at 15:39
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    $\begingroup$ Thank you for showing your answer back! Your previous answer was appreciated, and this is even more. Also, I agree with you and @TeresaLisbon that they should've shared some material about it, but I just find it weird that in page 110, Section 57 [The Sigma Notation], they recommended seeing college algebra books about the sum of an arithmetic progression, but not in the problem that I asked. $\endgroup$
    – soupless
    Commented Mar 10, 2021 at 15:58
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    $\begingroup$ @soupless Yes, a very odd textbook. Listen, I've seen this happen : some textbooks, in order to improve accessibility by engineers and beginners, just mention arguments in a flimsy way and then say it can be justified in rigorous mathematics. That kind of a textbook can be used by engineers and whoever needs it, but should not be used by anybody who wants rigorous justifications and is doing a course in rigorous mathematics. $\endgroup$ Commented Mar 10, 2021 at 16:11

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