Science:Math Exam Resources/Courses/MATH103/April 2013/Question 06 (c)/Statement

Light Show on the Dance Floor

A laser is mounted 2m above the floor (see sketch) and its beam is directed along a straight line on the dance floor. For visual effects, the angle $\theta$ changes periodically and randomly. The angle $\theta$ is chosen uniformly in the interval $(-{\frac {\pi }{4}},{\frac {\pi }{4}})$, that is, the probability density function (pdf) is $p(\theta )={\frac {2}{\pi }}$ for $-{\frac {\pi }{4}}<\theta <{\frac {\pi }{4}}$ and zero elsewhere.

The mean x-coordinate $\displaystyle {\overline {x}}$ of the laser point is given by $\displaystyle {\overline {x}}=\int _{a}^{b}xp(x)\,dx$ where a corresponds to the x value for $\displaystyle \theta ={\frac {\pi }{4}}$ and b corresponds to the x value for $\displaystyle \theta ={\frac {-\pi }{4}}$. Evaluate this integral (or state that it diverges).

NOTE: There was an error in the question as written on the exam and has been fixed on the wiki.