Solve $\int_0^2\sqrt{x} \, d\sqrt{x}\overset{?}{=}$

Method 1

$$\int_0^2 \sqrt{x} d\sqrt{x}$$

$$=\int_{\color{brown}{\sqrt{x}=0}}^{\color{brown}{\sqrt{x}=2}} \sqrt{x} d\color{brown}{\sqrt{x}}$$

$$= \left[\frac{(\sqrt{x})^2}{2}\right]_0^2$$

$$=\left[\frac{(2)^2}{2}\right]-\left[\frac{(0)^2}{2}\right]=2$$

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Method 2

$$\int_0^2 \sqrt{x} d\sqrt{x}$$

$$\int_{\color{brown}{\sqrt{x}=0}}^{\color{brown}{\sqrt{x}=2}} \sqrt{x} d\color{brown}{\sqrt{x}}$$

$$\sqrt{x}=t \implies \frac{d\sqrt{x}}{dt}=1 \implies d\sqrt{x}=dt$$

$$\bbox[2pt, border: 2pt green solid]{\sqrt{x}=t=2, \sqrt{x}=t=0}$$

$$=\int_{t=0}^{t=2} t dt=\frac{t^2}{2}\Big]_0^2=2$$

Using a few academic sources:

http://www.math.unl.edu/~jorr1/math826/RiemannStieltjes.pdf (theorem 6.7.8)

http://ocw.nctu.edu.tw/upload/classbfs1209122139184046.pdf (theorem E6)

These both state that:

$$\int_a^b f(x) d\alpha(x)=\int_a^b f(x) \alpha'(x) dx$$

no changing of limits of integration, agreeing with your answer of $1$. You DO NOT change limits with a Stieljes integral.

However lets go to the definition of Stieltjes integral

Lets compute

$$\begin{align*} \lim\limits_{n\to\infty} \sum\limits_{i=0}^{n-1} \sqrt{\frac{2i}{n}}\left(\sqrt{\frac{2(i+1)}{n}}-\sqrt{\frac{2i}{n}}\right)&=\lim\limits_{n\to\infty} \frac{2}{n}\sum\limits_{i=0}^{n-1} \sqrt{i^2+i}-i \\ \end{align*}$$

I tried to find a clever way of summing this (I couldn't), but pick a large number, say $n=1,000,001$ and using Wolfram Alpha

http://www.wolframalpha.com/input/?i=sum+sqrt%28k%5E2%2Bk%29-k+from+0+to+1000000

We get the approximation

$$\approx\frac{2}{1,000,001}\times 499,999$$

This is needless to say, pretty close to $1$.

Also, another argument look at the first source, theorem 6.7.6 integration by parts. This says that

$$\int_a^b f dg +\int_a^b g df=g(b)f(b)-g(a)f(a)$$

Since $f=g$, we get that:

$$2\int_0^2 \sqrt{x} d\sqrt{x}=\sqrt{2}\sqrt{2}-\sqrt{0}\sqrt{0}=2$$

That is,

$$\int_0^2 \sqrt{x} d\sqrt{x}=1$$

Edit: I wanted to add why there is the confusion about changing limits. I want to reiterate. With a Stieltjes integral:

$$\int_a^b f(x) d\alpha(x)=\int_a^b f(x) \alpha'(x) dx\neq \int_{\alpha^{-1}(a)}^{\alpha^{-1}(b)} f(x) \alpha'(x) dx$$

And in both the left and middle terms, you are integrating with limits of $x$. Not $\alpha(x)$, but in fact $x$. This is why things like $\int_{\sqrt{x}=0}^{\sqrt{x}=2}\sqrt{x}d\sqrt{x}$ in the other answer are nonsense.

The issue here is that this looks like u substitution from freshman calculus. $u$ substitution has the following form:

$$\int_{a}^{b} f(\alpha(x))\alpha'(x)dx=\int_{\alpha(a)}^{\alpha(b)}f(u) du$$

However, with Stieltjes integration there is an analogous formula (check theorem E9 in the second reference).

$$\int_a^b f(g(x))d\alpha(g(x))=\int_{g(a)}^{g(b)} f(u) d\alpha(u)$$

So in this case, let $f(x)=\operatorname{id}(x)=\alpha(x)$, $g(x)=\sqrt{x}$. Then:

$$\int_0^2 \operatorname{id}(\sqrt{x})d(\operatorname{id}(\sqrt{x}))=\int_0^{\sqrt{2}}\operatorname{id}(u)d(\operatorname{id}(u))=\int_0^{\sqrt{2}}u du=1$$

You need to change the limits of integration - if $\sqrt x$ goes from 0 to 2 then $x$ goes from 0 to 4.