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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?

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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 \( n \to \infty \).

Find a, b, and f(x) such that the limit lim (n→∞) Σ (i=1 to n) (8i^6 / n^7) is equal to ∫[a to b] f(x) dx. Enter them as variables a, b, and f respectively.

Explore how to convert the Riemann sum expression lim (n→∞) Σ (i=1 to n) (8i^6 / n^7) to a definite integral with specified limits and integrand. Learn about Riemann sums, definite integrals, and their applications in calculus.

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