[Solved] EE6506 Assignment 2

1. According to the Gauss quadrature rule, an integral is expressed as a weighted sum of n terms in the form: ⁿ₌₁ w_i f(x_i), where w_i, w₂, w_n are unknown coefficients known as weights and x₁, x₂ x_n are the discretization points. Construct a Gauss quadrature rule and use it to evaluate the following integral:^ex^x₂ dx. In addition:
   • Plot the function in the interval [0.1 4.9].
   • Show the development of the quadrature rule, noting that the limits asked above are non-standard. Bonus points if you derive the nodes and weights using symbolic math in MATLAB.
   • Study the accuracy of your rule as a function of the number of quadrature points, while comparing your output with the integral command in MATLAB for the same evaluation.
2. A thin metallic cylinder of length L and radius a, along the y axis as shown in the figure. The electrostatic potential on the cylinder is given as 1V. Using point matching and expressing the surface charge density _s in terms of a line charge density _l, in turn expressed via a pulse basis expansion:

N _l ⁼ a_ng_n(y), _l = 2a_s,

n=1

solve the following:

• find and plot the surface charge density on the cylinder _s,
• plot the potential V over the surface of the sphere with radius 10m.

Assume cylinder length L = 1m, and radius a = 0.01m. It is recommended to at least use a 3-point Gauss-quadrature rule to evaluate the integrals. Bonus points if you can justify the choice of the number of quadrature points; also if you can re-use some code from Q1 here.