To solve this problem, we are given the expression:

$\lim_{{n \to \infty}} \sum_{{i=1}}^n \frac{8 i^6}{n^7}$

and asked to find $a$, $b$, and $f(x)$ such that this limit is equal to the integral:

$\int_a^b f(x) \, dx$

This problem is formulated in terms of Riemann sums. To interpret it as an integral, let’s proceed with these steps:

Step 1: Recognize the Structure of the Riemann Sum

A Riemann sum has the general form:

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

where $\Delta x = \frac{b - a}{n}$, $x_i = a + i \Delta x$, and $f(x_i)$ represents the function being integrated.

Step 2: Identify $\Delta x$, $f(x_i)$, and the Limits $a$ and $b$

In the given sum: $\frac{8 i^6}{n^7}$

we can see that:

1. $\Delta x = \frac{1}{n}$.
2. $f(x_i)$ seems to correspond to the function evaluated at $x_i$, with $f(x_i) = 8 \left( \frac{i}{n} \right)^6$.
3. The term $\frac{i}{n}$ is a common substitution for $x$ in Riemann sums as $n \to \infty$.

Step 3: Rewrite $f(x)$

From $f(x_i) = 8 \left( \frac{i}{n} \right)^6$, we deduce that the function $f(x)$ is:

$f(x) = 8x^6$

Step 4: Determine the Interval $[a, b]$

Since $\frac{i}{n}$ goes from $\frac{1}{n}$ (when $i = 1$) to $1$ (when $i = n$), in the limit as $n \to \infty$, the interval for $x$ will be $[0, 1]$.

Thus, we have:

• $a = 0$
• $b = 1$

Conclusion

The values of $a$, $b$, and $f(x)$ are:

• $a = 0$
• $b = 1$
• $f(x) = 8x^6$

Final Answer

The integral representation is:

$\int_0^1 8x^6 \, dx$

Would you like more details on the integration process or have any questions?

Here are some related questions for further practice:

1. How would you interpret a sum with a different exponent, like $i^5$ instead of $i^6$?
2. What if the function inside the sum was $\frac{i}{n^2}$ instead? How would $f(x)$ and the interval change?
3. How do you compute the integral $\int_0^1 8x^6 \, dx$ to get a numeric answer?
4. What if the interval was $[1, 2]$ instead of $[0, 1]$? How would that affect $a$, $b$, and $f(x)$?
5. How does the Riemann sum change if $f(x)$ includes higher or lower powers of $x$?

Tip: When working with Riemann sums, the term $\frac{i}{n}$ often represents a variable $x$ that transitions to a continuous range in the limit as $n \to \infty$.