Application to the case $N=2$

The exact answer is $2/3$. Using $N=2$ with the above two weights and mesh points we get $$\begin{equation*} I=\int_{-1}^1x^2dx =\sum_{i=0}^{1}\omega_ix_i^2=\frac{1}{3}+\frac{1}{3}=\frac{2}{3}, \end{equation*}$$ the exact answer!

If we were to emply the trapezoidal rule we would get $$\begin{equation*} I=\int_{-1}^1x^2dx =\frac{b-a}{2}\left((a)^2+(b)^2\right)/2= \frac{1-(-1)}{2}\left((-1)^2+(1)^2\right)/2=1! \end{equation*}$$ With just two points we can calculate exactly the integral for a second-order polynomial since our methods approximates the exact function with higher order polynomial. How many points do you need with the trapezoidal rule in order to achieve a similar accuracy?

• «
• 1
• ...
• 52
• 53
• 54
• 55
• 56
• 57
• 58
• 59
• 60
• 61
• 62
• 63
• 64
• 65
• 66
• 67
• 68
• 69
• ...
• 95
• »