How to express a limit of Reimann sums as a definite integral

I need help understanding how to go from a limit of sums to a definite integral. I think I can go from integral to limit, but when I try to go from limit to integral I don't know how to tell what a and b are...


Anyway, this is the problem I have to do. But I'm going to need to do lots more so please explain how it is done! Thank you :)


Express the limit of sums as a definite integral, and use it to evaluate the limit.

limit as n -> infinity of the sum of (2+((4i^2)/(n^2))) (1/n) and under the sigma, i=1. On top of the sigma is n

So obviously 1/n = width of the rectangles = delta x
But like I said I have trouble deciphering the rest. Any help?

With Δx = 1/n, taking a = 0 yields b = 1 (because Δx = (b - a)/n).

Next, 2 + 4(i/n)^2 = 2 + 4(iΔx)^2 <---> f(x) = 2 + 4x^2.

So, this Riemann sum equals
∫(x = 0 to 1) (2 + 4x^2) dx = (2x + 4x^3/3) {for x = 0 to 1} = 10/3.

I hope this helps!