Answer each question with a minimum of 70 words per question: Compare and contrast adaptive quadrature and Gaussian quadrature Discuss the pros and cons of the Newton-Cotes formulae and Romberg...

Adaptive quadrature :
1)
    Adaptive quadrature
    Gaussian quadrature
    Definition :
In applied mathematics, adaptive quadrature is a process in which the integral of a function f(x) is approximated using static quadrature rules on adaptively refined subintervals of the integration domain. Generally, adaptive algorithms are just as efficient and effective as traditional algorithms for "well-behaved" integrands, but are also effective for "badly-behaved" integrands for which traditional algorithms fail.
    Definition :
In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the points xi​ and weights wi for i =1,2,...n
    Quadrature rules :
The quadrature rules generally have the form
Qn =
å
=
n
i
xi
wif
0
)
(
~
ò
b
a
dx
x
f
)
(
Where the nodes xi​ and weights wi are generally pre-computed.
    Quadrature rules :
The domain of integration for such a rule is conventionally taken as [−1, 1], so the rule is stated...