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Express the limit $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(4\left(x_{i}^{}\right)^{5}-5\left(x_{i}^{}\right)^{3}\right) \Delta x_{i}$ over $[5,8]$ as an integral.
Provide $a, b$ and $f(x)$ in the expression $\int_{a}^{b} f(x) d x$.
\[
a=\square, b=\square, f(x)=
\]
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Final Answer: $ \boxed{a=5, b=8, f(x)= 4x^{5} - 5x^{3}} $

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Step 1 ：The given limit is a Riemann sum for the integral of a function over the interval [5,8]. The function is given by the expression inside the sum, and the limits of integration are the limits of the interval.

Step 2 ：Therefore, we can say that a=5, b=8, and f(x) = 4x^5 - 5x^3.

Step 3 ：Final Answer: $ \boxed{a=5, b=8, f(x)= 4x^{5} - 5x^{3}} $

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