Integral of $1 / \sqrt x$ using Limits

Solution 1:

If the interval is positive and avoids $0$, the function is continuous, and thus integrable. Since it is integrable, any refinements of partitions will approach the correct answer. We then would like to set up a partition $a,aq,aq^2,\cdots$ where $q=(b/a)^{1/N}$ for an order $N$ partition. Note that Riemann’s integral does not require partitions to be equally spaced; he only requires that in refinement, the largest spacing (the “mesh”) goes to zero, which this geometric partition does. Also note that this geometric partition runs from $a\to b$ which is of course also necessary! Your sum with equal spacing would also work, but... it is extremely difficult to evaluate, whereas this one is easier as we shall soon see.

Note also that this method will integrate $x^a$ for any rational power of $a\neq-1$; do this as an exercise, maybe!

Our partial sums are:

$$\begin{align}\sum_{n=0}^{N-1}(aq^n)^{-1/2}(aq^{n+1}-aq^n)&=\sum_{n=0}^{N-1}(aq^n)^{-1/2}aq^n(q-1)\\&=\sum_{n=0}^{N-1}a^{1/2}q^{n/2}(q-1)\\&=a^{1/2}(q-1)\sum_{n=0}^{N-1}q^{n/2}\end{align}$$

Hopefully you are familiar with the geometric series:

$$a^{1/2}(q-1)\sum_{n=0}^{N-1}q^{n/2}=a^{1/2}(q-1)\frac{q^{N/2}-1}{q^{1/2}-1}=a^{1/2}(q-1)\frac{(b/a)^{1/2}-1}{q^{1/2}-1}$$

Which resolves to:

$$a^{1/2}((b/a)^{1/2}-1)\frac{q-1}{q^{1/2}-1}=(b^{1/2}-a^{1/2})\cdot\frac{q-1}{q^{1/2}-1}$$

I have for the last fraction (difference of two squares):

$$\frac{q-1}{q^{1/2}-1}=q^{1/2}+1\to2,\,N\to\infty$$

Since $\lim_{N\to\infty}(b/a)^{1/N}=1$, assuming $(b/a)\gt0$ which we have done. If you want to generalise this, put the power (as a rational) as $r/s$, and put $\tau=q^{1/s}$; you'll get two geometric series, and using a similar argument you'll get the right answer. To generalise this to irrational powers $\neq-1$, you use the continuity of exponentiation and the mean value theorem for integration to approximate it closer and closer with integrals of rational power; you arrive at the power law for integration at the end, and together with the fundamental theorem of calculus you arrive at a formal derivation of the power law of differentiation for all non-zero real powers! I found this personally to be a very instructive from-first-principles derivation: this is all due to Courant's Differential and Integral Calculus, a very old but pretty good textbook.

The final answer is then:

$$\int_a^b\frac{1}{\sqrt{x}}\,\mathrm{d}x=\lim_{N\to\infty}\sum_{n=0}^{N-1}\frac{1}{\sqrt{aq^n}}(aq^{n+1}-aq^n)=(b^{1/2}-a^{1/2})\cdot\lim_{N\to\infty}\frac{q-1}{q^{1/2}-1}=2(b^{1/2}-a^{1/2})$$

As required.