Numerical integration is motivated by the idea that not every integral can be found, but can be approximated. It won’t be the true value, but assuming we know what tolerance we want, we can calculate to that tolerance (which can be pretty damn good).
We explored Riemann sums (left, right, and mid endpoints) and the trapezium rule as a means of approximating definite integrals.
Why study this?
The emphasis on Riemann sums and approximating definite integrals was really unpopular with peers in first-year. Other schools had a fairly procedural way of teaching calculus (the way I learned it in high school) but we insisted on a more conceptual way. Tools like Wolfram let us solve integrals really easily now — but our work as engineers means sometimes we have to approximate them, and understanding how to do this effectively leads to solid foundations for future mathematics.1