Answer is: option4

Solution:

We are given a definite integral and asked to find which Riemann sum approximation converges to it:

\[ \int_1^3 x^3 \, dx \]

The definite integral

\[ \int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x \]

For a uniform partition of [1, 3] into \( n \) subintervals:

1. \( a = 1, \, b = 3 \), so the interval length is \( b - a = 2 \)
2. \( \Delta x = \frac{b-a}{n} = \frac{2}{n} \)
3. General form of a sample point: \( x_i^* = a + i \cdot \Delta x = 1 + \frac{2i}{n} \)

We are integrating \( f(x) = x^3 \), so the Riemann sum becomes:

\[ \sum_{i=1}^n \left( 1 + \frac{2i}{n} \right)^3 \cdot \frac{2}{n} \]

Now match this with the options.

(D): \[ \lim_{n \to \infty} \sum_{i=1}^n \left( 1 + \frac{2i}{n} \right)^3 \cdot \frac{2}{n} \]