Writing a Limit as a Definite Integral In Exercises 9-12 ,...

Writing a Limit as a Definite Integral In Exercises $9-12$ , write the limit as a definite integral on the interval $[a, b],$ where $c_{i}$ is any point in the ith subinterval. $$ \text {Limit} \quad \text {Interval} $$ $$ \lim _{\|\Delta\| \rightarrow 0} \sum_{i=1}^{n}\left(\frac{3}{c_{i}^{2}}\right) \Delta x_{i} $$

Writing a Limit as a Definite Integral In Exercises $9-12$ , write the limit as a definite integral on the interval $[a, b],$ where $c_{i}$ is any point in the ith subinterval.
$$
\text {Limit} \quad \text {Interval}
$$
$$
\lim _{\|\Delta\| \rightarrow 0} \sum_{i=1}^{n}\left(\frac{3}{c_{i}^{2}}\right) \Delta x_{i}
$$

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Key Concepts

Definite Integral

A definite integral represents the accumulation of quantities and is defined as the limit of a Riemann sum as the number of subintervals tends to infinity and their maximum width approaches zero. It is used to calculate areas under curves, among other applications, across a specified interval [a, b].

Riemann Sum

A Riemann sum approximates the area under a curve by summing up the products of function values at selected sample points within subintervals and the corresponding widths of those subintervals. This is fundamental in transitioning from discrete sums to the continuous nature of integration.

Partition of an Interval

Partitioning an interval involves breaking the interval [a, b] into smaller subintervals, each of which has a width ?x. This division is essential in calculating Riemann sums and forms the basis of defining the definite integral, as it allows for the approximation of the area or other cumulative quantities.

Limit Process in Integration

The limit process involves taking the limit as the norm of the partition (the width of the largest subinterval) approaches zero. This ensures that the Riemann sum converges to the exact value of the definite integral, providing a precise calculation of the accumulated quantity over the interval.

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