Here are several problems solvable using Adaptive Simpson’s Method:
- HDU 1724 - Ellipse
- ICPC Live Archive 3485 - Bridge (solvable without Simpson’s Method)
- SGU 217 - Two Cylinders (on CodeForces)
- ICPC Live Archive 5096 - Volume (also solvable without Simpson’s Method)
- ICPC Live Archive 4330 - Timer (once again solvable without Simpson’s Method)
There aren’t many problems involving numerical integrals indeed. Also, you can find such problems on CodeChef by searching with tags Integrals here.