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. $$ \text {Limit} \quad \text {Interval} $$ $$ \lim _{\|\Delta\| \rightarrow 0} \sum_{i=1}^{n}\left(3 c_{i}+10\right) \Delta x_{i} \quad[-1,5] $$

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.
$$
\text {Limit} \quad \text {Interval}
$$
$$
\lim _{\|\Delta\| \rightarrow 0} \sum_{i=1}^{n}\left(3 c_{i}+10\right) \Delta x_{i} \quad[-1,5]
$$

Key Concepts

Limiting Process in Integration

The limiting process in integration refers to the approach of taking the limit of a Riemann sum as the norm of the partition (i.e., the width of the largest subinterval) tends to zero. This process transforms the approximation of the sum into an exact calculation, resulting in the definition of the definite integral.

Riemann Sum

A Riemann sum is a method for approximating the area under a curve by dividing the interval into smaller subintervals, choosing a sample point in each subinterval, and summing the product of the function’s value at each sample point and the width of the subinterval. This approach lays the foundation for the transition from discrete sums to continuous integration.

Definite Integral

A definite integral represents the exact accumulation of quantities, such as area under a curve, over a specific interval. It is defined as the limit of a Riemann sum as the maximum subinterval width approaches zero, providing an exact value for the aggregate effect of a function over an interval.

Partitioning of an Interval

Partitioning an interval involves dividing it into a finite number of subintervals which are used to approximate the behavior of a function over the whole interval. This concept is essential for constructing Riemann sums and understanding how discrete approximations converge to continuous quantities.

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