Description
1. According to the Gauss quadrature rule, an integral is expressed as a weighted sum of n terms
in the form: R b
a
f(x)dx ≈ ∑
n
i=1 wi
f(xi), where wi
, w2, · · · wn are unknown coefficients known
as weights and x1, x2 · · · xn are the discretization points. Construct a Gauss quadrature rule
and use it to evaluate the following integral: R 4
1
e
x
√
x
2
dx. In addition:
(a) Plot the function in the interval [0.1 – 4.9].
(b) 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.
(c) 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 ρs
in terms of a line charge density ρl
, in turn expressed via a pulse
basis expansion:
ρl =
N
∑
n=1
angn(y), ρl = 2πaρs
,
solve the following:
(a) find and plot the surface charge density on the cylinder ρs
,
(b) 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.
