# Approximating this integral

1. May 24, 2007

### e12514

If g maps a closed bounded subset of R to R^n
g : [a,b] -> R^n
and g is continuous,

can the definite integral
(integral from a to b) g(t) dt

be approximated by
(SUM from 0 to n-1 ) ( g(t) (b-a)/n ) ?
(because as taking n->oo gives the integral?)

If so, what are the steps needed to go from the integral to the sum (obviously it's not valid just to claim they're eqaul)?
Or if not, what's the finite sum that approximates that integral?

And also, how do we actually rigourously show that the approximation sum converges to a limit equal to the integral?

2. May 25, 2007

### AiRAVATA

Isn't that almost a Riemann sum?

(if instead of taking $\sum g(t)\frac{b-a}{n}$ you take $\sum g(\xi_n)\frac{b-a}{n}$ where $\xi_n$ is in each interval of the partition, then, in the limit, is exactly the Riemann sum.)