I have the following question can anyone help me on this?
I am studying methods to approximate integrals such as composite second order or higher Gauss quadrature rule and Monte Carlo method.
I have the following integrals:
1- $\int_{2050}^{-400} (\frac{3x^9}{11}+4x^5-200x^2-\pi*x^4-23) dx$
and
2- $\int_{D} ($$\prod_{i=3}^{32} \frac {x_i-x_{i-2} }{4}- sin(x_{20}x_{8}x_{64}) $$ dx$ D is contained in $R^{32}$ $ {(x_1,x_2,x_3,x_4...x_{32}): x^2_1+x^2_3+x^4_4+x^4_2+x^2_{31}+x^4_{32} } < 110 $
I am wondering which method is better to solve each integral by using Gauss quadrature or Monte Carlo and I I think that for the first one is better to use Gauss quadrature with order 5 (n=5) because the degree of precisions is 2n-1 and the it is a function of degree 9.
For the second one is better to use Monte Carlo because the dimension is too high.
Am I thinking correctly? Can anyone give me some help on this?
Thanks