Consider a real-valued function defined on a closed interval. One of the fundamental objectives in calculus is to determine the area under the graph of such a function. When an exact computation is not readily available, this area can be estimated by dividing the interval into a finite number of equal subintervals. Each subinterval corresponds to a rectangle whose width is the length of the subinterval and whose height is determined by the value of the function at a selected point within that subinterval.
For each subinterval, an arbitrary point is chosen, and the function value at that point is used as the height of the rectangle. The area of each rectangle is obtained by multiplying its height by its width. Adding the areas of all rectangles provides an approximation of the total area under the curve. This approximation is known as a Riemann sum. As the number of subintervals increases and their widths decrease, the approximation generally becomes more accurate, reflecting the shape of the curve more closely.
The definite integral refines the idea of a Riemann sum by considering what happens when the partition of the interval becomes infinitely fine. The definite integral of a function over a given interval is defined as the value approached by the Riemann sums as the number of rectangles grows without bound and the width of each rectangle becomes negligibly small, provided that this limiting value exists. In this sense, the definite integral represents the exact area under the curve over the specified interval.
A practical illustration of this concept arises in the study of motion. Consider a car traveling along a straight path with a velocity that varies over time. By dividing the total travel time into equal segments and using the velocity at a chosen moment within each segment, the distance traveled during that segment can be estimated. Summing these distances yields a Riemann sum that approximates the total distance traveled. When the velocity remains positive throughout the interval, the definite integral of the velocity function corresponds to the total distance covered by the car.
Consider a function over a closed interval. To approximate the area under the curve, divide the interval into equal subintervals, forming rectangles of equal width.
Select an arbitrary point within the subinterval and use the function value there as the rectangle’s height. Multiplying the height by the width gives the area of that rectangle. The total area under the curve is approximated by summing the areas of all rectangles and is called a Riemann sum.
The definite integral of a function over a given interval is defined as the limit of its Riemann sum as the number of subintervals approaches infinity and the width of each rectangle becomes infinitesimally small, provided that this limit exists.
For example, consider a car traveling with a varying velocity. Divide the total travel time into equal time intervals.
In each interval, select an arbitrary time point and take the corresponding velocity. Multiply the velocity by the interval width to estimate the distance traveled.
The sum of these forms a Riemann sum, and in this example, the area under the curve is physically interpreted as the distance traveled by the car. When the velocity remains positive, the total distance is found by integrating the velocity function within the limits.
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