Write the limit as a definite integral on the interval [a,...

Write the limit as a definite integral on the interval $[a, b]$, where $c_{i}$ is any point in the $i$ th subinterval.
Limit
$$
\lim _{\|\Delta\| \rightarrow 0} \sum_{i=1}^{n}\left(3 c_{i}+10\right) \Delta x_{i}
$$
Interval
$$
[-1,5]
$$

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Key Concepts

Riemann Sum

A Riemann sum is a method of approximating the value of a definite integral by partitioning the interval into subintervals, selecting a sample point in each, and summing the product of the function value at each point and the width of the corresponding subinterval. This sum approaches the integral as the partition becomes infinitesimally fine.

Partition of an Interval

Partitioning an interval involves dividing it into smaller subintervals, typically with a variable width. The finer the partition (i.e., the smaller the maximum width), the more accurate the approximation provided by the Riemann sum, leading to the exact value of the integral in the limit.

Definite Integral

A definite integral calculates the accumulation of a quantity, such as area under a curve, over a specified interval by taking the limit of a sum. It is defined as the limit of a Riemann sum as the maximum subinterval length approaches zero, ensuring that the approximation becomes exact.

Limit Process in Integration

The limit process in integration refers to the concept of taking the limit as the maximum subinterval length (norm of the partition) approaches zero. This process assures that the approximation using Riemann sums transitions seamlessly into the exact value given by the definite integral.

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