Express the given limit of a Riemann sum as a definite...

Express the given limit of a Riemann sum as a definite integral and then evaluate the integral.
$$\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(8\left(1+\frac{i}{n}\right)-8\right) \cdot \frac{1}{n}$$

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Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration. It states that if a function is continuous over an interval, then its definite integral over that interval can be evaluated using its antiderivative. This theorem is crucial in converting the limit of a Riemann sum into a computable integral and then evaluating it efficiently.

Riemann Sum

A Riemann sum is a method for approximating the area under a curve by dividing the interval into small subintervals and summing the product of the function's value at representative points and the width of each subinterval. It forms the foundation for defining and understanding definite integrals as the limit of these sums when the number of subintervals approaches infinity.

Definite Integral

A definite integral represents the exact area under a curve between two specified limits. It is defined as the limit of the Riemann sum as the number of subintervals increases indefinitely, capturing the cumulative sum of infinitesimally small slices and providing a precise way to calculate accumulated quantities.

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