The Average Wait

A delivery app tracks how long a courier takes to reach a restaurant: a continuous time $X$, in minutes, that always lands somewhere between $a$ and $b$. The data team has worked out the shape of its density curve, a function shape(x) that is never negative, but they never scaled it. The area under it is generally not 1.

The average of a continuous quantity is an integral: every possible value, weighted by its density, added up.

Here $f$ has to be a genuine density curve: never negative, and enclosing a total area of exactly 1.

Task: write density_mean(shape, a, b, n) that estimates the average of $X$ with rectangles and returns it rounded to 4 decimal places.

• shape is a Python function that takes one number and returns one number: shape(2.5) is the height of the curve at $x = 2.5$.
• Split $[a, b]$ into n rectangles of equal width.
• Take each rectangle's height at the middle of its strip, not at either edge.
• Use these same n slices for every area you estimate.

For example, for the shape $x(4 - x)^2$ on $[0, 4]$ with 4 rectangles, the answer is 1.5568.