Question

How do I convert `lim_(n->oo) sum_(i=0)^n (12n)/(16n^2 + 9i^2)` into a definite integral?

I'm new when it comes to Riemann sums. I also have trouble with finding `Delta x_i`.

Answer 1

`arctan(3/4)`

Explanation:

`lim_(n->oo) sum_(i=0)^n (12n)/(16n^2 + 9i^2) = lim_(n->oo) sum_(i=0)^n (12/(16 + 9(i/n)^2))1/n`

and calling now `xi = i/n` and `d xi = 1/n`

`lim_(n->oo) sum_(i=0)^n (12/(16 + 9(i/n)^2))1/n=int_0^1 12/(16+9 xi^2)d xi = arctan(3/4)`