I have this question Derive an open two-point Newton-Cotes quadrature rule for the interval $[a,b]$.
I need to find what the resulting weights and nodes are. What is the degree of the resulting rule?
Here is my work so far: I know that for data points that are open, $$ x_i = a + i\cdot\frac{b-a}{n+1} $$
so i have $x_1 = a + \frac{b-a}{2}$ and $x_2 = a + 2\frac{b-a}{3}$
how do i find the weights, nodes and degrees?
i think for the weights i have $$ \int_{a}^{b} 1 dx $$ and $$ \int_{a}^{b} x dx $$
but i do not know how to solve for them.
Actually, your points are $x_1=a+1\cdot\frac{b-a}3=\frac{2a+b}3$ and $x_2=a+2\cdot\frac{b-a}3=\frac{a+2b}3$
Now determine weights $w_1,w_2$ so that $$ w_1f(x_1)+w_2f(x_2)=\int_a^b f(x)\,dx $$ for $f(x)=1$ and $f(x)=x$, and if possible, also for $f(x)=x^2$.