Hi,
I am a bit confused by the the meaning of order in approximating polynomials. I though a constant is a zero order, a line is order one, a quadratic function is of second order and so on. In a simple rectangular approximation we do [imath]\int{f(x)} = \sum_{n=1}^{\infty} f_n dx[/imath]. In the trapezoidal method we do [math]\int{f(x)} = \sum_{n=1}^{\infty} \frac{h}{2}(f_n + f_{n+1}) dx[/math]. Now, it is said that in the rectengular method we integrate 0 order polynomial (a constant) to get an integral which is of order one. Here is my first question, how can the integral be an integral of order one? Likewise in the trapezoid method - it is said that we integrate a first order polynomial (a line) to get an integral of second order. So why in the average of two consecutive points in the trapezoid method considered a line? And more importantly why is the resulting integral second order? After all it is the same plain multiplication as it was in the rectengular case, just one of the arguments in an average of two points instead of a single one.
Thanks!
I am a bit confused by the the meaning of order in approximating polynomials. I though a constant is a zero order, a line is order one, a quadratic function is of second order and so on. In a simple rectangular approximation we do [imath]\int{f(x)} = \sum_{n=1}^{\infty} f_n dx[/imath]. In the trapezoidal method we do [math]\int{f(x)} = \sum_{n=1}^{\infty} \frac{h}{2}(f_n + f_{n+1}) dx[/math]. Now, it is said that in the rectengular method we integrate 0 order polynomial (a constant) to get an integral which is of order one. Here is my first question, how can the integral be an integral of order one? Likewise in the trapezoid method - it is said that we integrate a first order polynomial (a line) to get an integral of second order. So why in the average of two consecutive points in the trapezoid method considered a line? And more importantly why is the resulting integral second order? After all it is the same plain multiplication as it was in the rectengular case, just one of the arguments in an average of two points instead of a single one.
Thanks!