The problem asks to express the definite integral $\int_0^3 (x-16) dx$ as a limit of Riemann sums, using right endpoints. We need to determine the correct Riemann sum representation from the given options.

1. Problem Description

The problem asks to express the definite integral $\int_0^3 (x-16) dx$ as a limit of Riemann sums, using right endpoints. We need to determine the correct Riemann sum representation from the given options.

2. Solution Steps

The general form of a Riemann sum for the integral $\int_a^b f(x) dx$ using $n$ rectangles is given by:

$\lim_{n \to \infty} \sum_{i=1}^n f(x_i) \Delta x$

where $\Delta x = \frac{b-a}{n}$ is the width of each rectangle, and $x_i$ is the right endpoint of the $i$-th rectangle.

In our case, $a = 0$, $b = 3$, and $f(x) = x - 16$.

So, $\Delta x = \frac{3-0}{n} = \frac{3}{n}$.

The right endpoints are given by $x_i = a + i \Delta x = 0 + i \frac{3}{n} = \frac{3i}{n}$.

Therefore, $f(x_i) = f\left(\frac{3i}{n}\right) = \frac{3i}{n} - 16$.

The Riemann sum is:

$\lim_{n \to \infty} \sum_{i=1}^n \left(\frac{3i}{n} - 16\right) \frac{3}{n} = \lim_{n \to \infty} \sum_{i=1}^n (x_i - 16) \Delta x$

where $x_i = \frac{3i}{n}$ and $\Delta x = \frac{3}{n}$. This corresponds to the interval $[0, 3]$.

Comparing the derived expression with the given options, we can see that option B and C have the correct interval. But we need to identify the correct term within the summation. Option C correctly has $x$ instead of $x_i$ inside the summation. It doesn't matter if $x$ or $x_i$ are used, as long as it is clear that we are evaluating the function at the right endpoints.

Option B and C are

$\lim_{n \to \infty} \sum_{i=1}^n (x_i - 16) \Delta x$ over the interval $[0, 3]$ where $x_i = \frac{3i}{n}$.

$\lim_{n \to \infty} \sum_{i=1}^n (x - 16) \Delta x$ over the interval $[0, 3]$.

Based on the above observations, the only suitable option is option B.

3. Final Answer

B. $\lim_{n\to\infty} \sum_{i=1}^n (x_i - 16) \Delta x$ over the interval $[0, 3]$