MATLAB exercise for average of dot products of unit vectors not matching textbook's calculus result.

Question

I'm working through Strang's Introduction to Linear Algebra. Problem Set 1.2, #34 states,

Using $v=randn(3,1)$ in MATLAB, create a random unit vector $u=\frac{v}{||v||}$. Using $V=randn(3,30)$ create 30 more random unit vectors $U_j$. What is the average size of the dot products $|u\cdot U_j|$? In calculus, the average is $\int_0^\pi |\cos\theta| d\theta / \pi = 2/\pi$.

Here's my program:

v=randn(3,1)
u=v/norm(v)
sum=0
k=30
for i=1:k
    V=randn(3,1)
    U=V/norm(V)
    dot_product=abs(dot(u,U));
    sum+=dot_product;
end
sum/k

I am getting an average near $1/2$, not $2/\pi$. Same tendency as I increase k tenfold or hundredfold.

Sample output of my program below. The unit vectors look correct. Is there something I'm misunderstanding?

v =

  -0.9049
   0.4694
  -1.0034

u =

  -0.6326
   0.3281
  -0.7015

sum = 0
k = 30
V =

   0.443107
   0.064042
   0.274244

U =

   0.8440
   0.1220
   0.5223

...
...
...

V =

  -0.6647
   0.5873
  -0.9252

U =

  -0.5186
   0.4582
  -0.7219

ans = 0.5125

Answer 1

The integral looks very much like the 2D version where the angle between the two vectors is uniformly distributed between $0$ and $\pi$. Indeed if you do the Matlab exercise with "rand(2,1)" you get close to $2/\pi$.

Looks like an error in the book.