Can Riemann Sums Approximate Integrals of Vector-Valued Functions?

[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?

 

[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.)