Following is exercise 18 from page 72 of the book "A multigrid Tutorial" written by William L. Briggs, Van Emden Henson, and Steve F. McCormick:
Let $u(x)=x^{m/2}$, where $m > -1$ is an integer, on $\Omega = [0,1]$, with grid spacing $h = \frac{1}{n}$. Let $u_i^h=x_i^{m/2}=(ih)^{m/2}$. Show that the continuous $L^2$ norm is $||u||_2=(\int_0^1x^{m/2})^{1/2}=\frac{1}{\sqrt{m+1}}$ while the corresponding discrete $L^2$ norm satisfies $||u^h||_h\overset{\mathrm{h\rightarrow0}}{=}\frac{1}{\sqrt{m+1}}$.
However, it seems to me that the integration $\int_0^1x^{m/2}$ should lead to $\frac{2}{m+2}$, which is different from the answer...
In addition, the definition of the discrete $L^2$ norm is $||u^h||_h=(h^d\sum_i(u_i^h)^2)^{1/2}$, could you please shed the light on me what technique shall I use to prove its limit is its corresponding continuous $L^2$ norm $||u||_2=(\int_\Omega u(x)\mathrm{d} x)^{1/2}$?
P.S. This is not a homework, I am self-studying this book.