Evaluate the limit and sketch the graph of the region whose...

Evaluate the limit and sketch the graph of the region whose area is represented by the limit.
$\lim _{\|\Delta\| 0} \sum_{i=1}^{n}\left(x_{i}-x_{i}^{2}\right) \Delta x$, where $x_{i}=i / n$ and $\Delta x=1 / n$

Key Concepts

Riemann Sum

A Riemann sum is a way to approximate the area under a curve by dividing the region into small subintervals, computing the function's value at specific sample points within these subintervals, and then summing the product of these values with the subinterval widths. As the number of subintervals increases (and their maximum width decreases), the Riemann sum approaches the exact area under the curve.

Definite Integral

The definite integral is the limit of the Riemann sums as the width of the subintervals approaches zero. It represents the exact area under a function’s curve between two specific bounds by summing up infinitely many infinitesimally small areas.

Limit Process

The limit process in this context involves taking the partition size ?x and letting it tend to zero. This ensures that the approximation provided by a Riemann sum becomes exact, yielding the definite integral of the function over the interval.

Area Under a Curve

The concept of the area under a curve involves calculating the region bounded by the graph of a function, the x-axis, and vertical lines corresponding to the interval's endpoints. This area can be determined by evaluating the definite integral of the function over the interval.

Graphing Functions

Graphing a function involves plotting the curve of the function over the domain of interest; in this case, it typically means drawing the curve corresponding to a function and visualizing the region of interest. This helps in understanding the behavior of the function and the area represented by the definite integral.

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