Bepaalde integraal

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.