Submit Answer [-/1 Points] DETAILS \[ \begin{array}{l} \lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{i^{2}}{n^{3}} ; a=1 \\ f(x)=\square \end{array} \]

Step 1 ：The problem is asking for the limit of the sum as n approaches infinity. This is a Riemann sum problem. The Riemann sum is an approximation of the definite integral of a function, and it is calculated as the sum of the areas of rectangles under the curve of the function. The width of each rectangle is the interval of the function divided by the number of rectangles (n), and the height of each rectangle is the value of the function at the right endpoint of the rectangle's base. As n approaches infinity, the width of each rectangle approaches 0, and the Riemann sum approaches the exact value of the definite integral.

Step 2 ：In this case, the function inside the sum is \( \frac{i^{2}}{n^{3}} \). The variable i is the index of summation, and it ranges from 1 to n. As n approaches infinity, \( \frac{i}{n} \) approaches 0, and the function inside the sum approaches 0. Therefore, the limit of the sum as n approaches infinity is the definite integral of the function from 0 to 1.

Step 3 ：To find the function f(x) that corresponds to the Riemann sum, we need to replace \( \frac{i}{n} \) with x in the function inside the sum. This gives us the function \( f(x) = x^{2} \).

Step 4 ：Now, we can calculate the definite integral of f(x) from 0 to 1 to find the limit of the sum as n approaches infinity.

Step 5 ：The definite integral of \( f(x) = x^{2} \) from 0 to 1 is \( \frac{1}{3} \).

Step 6 ：Final Answer: The limit of the sum as n approaches infinity is \( \boxed{\frac{1}{3}} \).