Homework #7

due: Fri April 26, 2024

Each student is responsible for doing their own work. You are welcome to discuss this on slack or in class, but when it comes to writing the code, you are expected to write it yourself (and not just copy from someone).

Note: using ChatGPT or similar AI tools to write your code is not allowed.

1. In our debugging experiment, we looked at the trapezoid rule for integration.

   Given a function, $f(x)$, defined on $[a, b]$, we divide the domain into $N$ slabs, and the integral is approximated as:

   $$I = \frac{1}{2} \Delta x \sum_{i=1}^N \left ( f(x_i) + f(x_{i+1}) \right )$$

   where $\Delta x$ is the width of a slab and $x_i$ and $x_{i+1}$ are the coordinates of the left and right edge of slab $i$.

   a. Write a C++ function to integrate a function $f(x)$ using the trapezoid rule. Your C++ function should take as input the number of slabs, $N$, and the integrand, $f(x)$, using std::function, as we saw in class.

   b. Write a second C++ function to do Monte Carlo integration. The interface will be similar to your trapezoid function, but now $N$ will be the number of random points.

   c. Write a program that integrates

   $$I = \int_{-5}^5 e^{-x^2} dx$$

   using both the trapezoid rule and Monte Carlo. Vary N (say from 8 to 1024, by doubling), and have your program output a table with: $N$, trapezoid answer, MC answer.

   d. Now do the same, but for the integral:

   $$I = \int_0^2 \sin^2 \left ( \frac{1}{x (2-x) + \epsilon} \right ) dx$$

   where $\epsilon \approx 10^{-12}$ is there to prevent dividing by zero.

   You might want to plot this integrand to see the structure.