# Exercise 2b: multiprocessing and map

Objective: introduce `map` and `Pool.map`.

In the `numerical_integration.py` file, we give Python code that calculates
the integral of a function in two different ways: numerically and analytically.

The given functions are `integrate` (numerical integration), `f` (the function
to integrate), and `F` (the analytical integral).

We want to check the precision of the numerical integration as a function of
the number of steps in the domain. To do this, we calculate and print the
relative differences between the analytic result and the numerical result
for different values of the number of steps.

**TASKS**:  
 0. Read `numerical_integration.py` and familiarize yourselves with the code.
 1. Update the `main` function so that it calculates the numerical error without
    any parallelization. You can use a for loop or `map`.
 2. Note the execution time for this serial implementation.
 3. Implement the parallel version using `multiprocessing.Pool`.
 4. Compare the timing for the parallel version with the serial time.
    What speed-up did you get?

**BONUS TASKS (very optional)**:  
 5. Implement a parallel version with threads (using `multiprocessing.pool.ThreadPool`).
 6. Time this version, and hypothetize about the result.