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Use Definition 2 to find an expression for the area under...

Key Concepts

Riemann Sum

A Riemann sum is a method for approximating the area under a curve by dividing the interval into small subintervals, calculating the function's value at specific sample points in each subinterval, and summing up the areas of the resulting rectangles. This approach forms the foundation of integral calculus by providing an approximation that becomes exact in the limit as the widths of the subintervals approach zero.

Definite Integral

The definite integral represents the exact area under a continuous curve over a specified interval. It is defined as the limit of the Riemann sums when the number of subintervals increases indefinitely and the maximum width of the subintervals tends to zero. This limit process converts the approximate sum of rectangular areas into the precise accumulation of area.

Partition of an Interval

Partitioning an interval involves dividing the interval of integration into a finite number of smaller subintervals. This is an essential step in forming a Riemann sum, as it enables the approximation of the area under the curve by considering the contribution from each smaller segment of the interval.

Limit Process

The limit process is central to the concept of integration; it is used to refine the approximation provided by the Riemann sums. By taking the limit as the number of subintervals approaches infinity (and the width of each subinterval approaches zero), the approximated area converges to the exact area under the curve represented by the definite integral.

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