Originally Posted by

**johng** Your original post asked a question about the distribution of a r.v. g(X) (of course X is a r.v.) where function g is monotonic. The function $g(x)=x^2$ is __not__ monotonic.

Let $f(x)={1\over \sqrt{2\pi}}\exp(-x^2/2)$, the density of a standard normal N(0,1) r.v. X

$$\int_0^{10}f(x)dx$$

This integral represents the __probability__ of the event $(0\leq X\leq 10)$, which of course must be a real number between 0 and 1. So it's definitely not 1.25. This integral (probability) is very close to .5

Now you seem to want the probability of $(0\leq X^2\leq10)$. This event is the same as $(-\sqrt{10}\leq X\leq \sqrt{10})$. So its probability is

$$\int_{-\sqrt{10}}^{\sqrt{10}}f(x)dx\approx0.998$$