A statement in the book "Differential and Integral Calculus, 6th ed" (Love and Rainville) 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]


*Why does it need a rigorous justification? Isn't the given solution enough?

 A: "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.
