Assignment $9-$ Integration

For this assignment, you will be computing magnetic fields using the following integral,

$B={\displaystyle \underset{-\frac{L}{2}}{\overset{\frac{L}{2}}{}}}[\frac{{}_{0}Ia}{4}\frac{dx}{(a^2+x^2{)}^{\frac{3}{2}}}]$

where a represents the distance away from a wire and $L$ represents the length of the wire, and ${}_0=1\mathrm{.}257{10}^{-6}$Tesla$\frac{m^2}{A}($the permeability of free space$).$ For all of your computations, let $I=1\mathrm{.}5$Amps,$L=10cm,$ and $a=3cm.$ For all of your computations, report the magnetic field in units of microteslas $(T).($Be sure to pay attention to all of the units!$)$

Use sympy to calculate this integral analytically and to determine the value of $B.$ Measure the computation time for the sympy calculation.

Use the quad function to calculate this integral numerically to determine the value of $B.$ Discuss the size of the numerical error for this computation. Measure the computation time for the quad calculation.

Create a function that works just like the quad function, but takes an extra input parameter $"$N$"$ and computes the integral using a Riemann sum of $N$ rectangles. Use your function to compute the integral for different values of $N.$ For each of the different values of $N,$ report both the result of the integral and the computation time. Discuss these results. Specifically, discuss how the results of the three different methods $($sympy$,$ quad, and Riemann sum$)$ compare to each other in terms of both the accuracy of the computation and the computation time.