Question

Which of the following options correctly expresses the limit as a definite integral?

1) (lim_{n o infty} sum_{i=1}^{n} f(x_i) Delta x)
2) (lim_{n o infty} rac{b-a}{n} sum_{i=1}^{n} f(x_i))
3) (lim_{n o infty} rac{1}{n} sum_{i=1}^{n} f(x_i))
4) (lim_{n o infty} rac{b-a}{n} sum_{i=1}^{n} f(x_i) Delta x)

Answer 1

4) lim (n -> ∞) [(b-a)/n * Σ(i=1 to n) f(xi) Δx] correctly expresses the limit as a definite integral

Explanation:

The correct option expressing the limit as a definite integral is option 4. It represents the limit as n approaches infinity of the Riemann sum [(b-a)/n * Σ(i=1 to n) f(xi) Δx]. This expression is a representation of the definite integral of the function f(x) over the interval [a, b] with respect to x, where Δx represents the width of each subinterval, and f(xi) is the value of the function at a sample point within each subinterval.

In the context of calculus, the limit of a Riemann sum as the number of subintervals (n) approaches infinity corresponds to the definite integral of a function over a given interval. The expression (b-a)/n represents the width of each subinterval, and as n becomes infinitely large, the sum approaches the integral.

In summary, option 4 correctly represents the limit as a definite integral and corresponds to the fundamental concept of calculus that connects the accumulation of quantities represented by the definite integral to the limit of Riemann sums as the number of subintervals becomes infinite.