Evaluate the limit by interpreting it as the limit of a Riemann sum of a function on the interval [a, b].
B)-
C)-

Question

Evaluate the limit by interpreting it as the limit of a Riemann sum of a function on the interval [a, b]. 
B)- 
C)-

Quizplus · Accepted Answer

We cannot determine the exact function and interval without more information, but we can interpret the given limit as a Riemann sum.

Recall that a Riemann sum is a sum of the form:

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

where $f(x)$ is a function on the interval $[a,b]$, $x_i$ is a point in the $i$th subinterval, and $\Delta x$ is the width of each subinterval. As the number of subintervals ($n$) increases, the Riemann sum approaches the definite integral:

$$\lim_{n	o\infty}\sum_{i=1}^n f(x_i)\Delta x = \int_a^b f(x) dx$$

In the given limit, we can interpret 11eaa3e5_5581_cec9_9f8f_c9e9405a2566_TB5971_11 as the value of a function at some point. Let this function be $f(x)$. Then we can rewrite the limit as:

$$\lim_{n	o\infty}\sum_{i=1}^n f(x_i)\Delta x = \lim_{n	o\infty}\left[f(x_1)\Delta x + \cdots + f(x_n)\Delta xight]$$

where $x_1=a$ and $x_n=b$. Notice that this is of the form of a Riemann sum. As $n$ gets larger and larger, the sum becomes a better and better approximation of the definite integral of $f(x)$ on the interval $[a,b]$.

The given limit is then equivalent to the definite integral of $f(x)$ on the interval $[a,b]$:

$$\lim_{n	o\infty}\sum_{i=1}^n f(x_i)\Delta x = \int_a^b f(x)dx$$

Therefore, the best choice is D, which represents the value of the definite integral.

Evaluate the Limit by Interpreting It as the Limit of a Riemann