Numerical Integration Quiz
You may solve this quiz using either Mathematica or Python. In both cases be sure to include not only the programs, but plots or values output by your programs. If needed you may need to handwrite and scan or type your results into a file to include in the ZIP file that you e-mail to me.
Part 1
When studying Fresnel Diffraction one encounters two integrals
These two Fresnel integrals define the Cornu Spiral, which is a graph of S(u) versus C(u). These two functions have the upper limit of the integration, u, as their independent variable. Numerically integrate S(u) and C(u) for u in the range -10 < u < 10 and then plot S(u) versus C(u) to get the Cornu Spiral.
Part 2
According to the Big Bang model of the universe, the age of the universe is given by the integral
where t0 is called the Hubble time, t0 = 13.4 Gyr. Ωm is the matter density parameter, and ΩΛ is the dark-energy density parameter. Write a program that takes Ωm and ΩΛ as inputs and computes the age of the universe. What is the age of the universe given the following conditions?
| Ωm | ΩΛ |
|---|---|
| 0.04 | 0.0 |
| 1.0 | 0.0 |
| 0.3 | 0.0 |
| 0.3 | 0.7 |
| 0.3 | 1.4 |
| 0.3 | 0.35 |