Definition:If a function defined on an interval and then a definite integral of a function on an interval is a number equal to limit of integral sum where f
The construction of integral sums for the example of determining the area of the curvilinear trapezoid
Let on the segment is set to an integral and continuous function
To determine the area of the curvilinear trapezoid (bounded curve axis and straight, and ), divide the cut points
on the parts selected on each of the obtained partial segments of an arbitrary point of the calculated values of the function at these points and form the sum where
This amount is equal to the sum of the areas of the shaded rectangles is called the integral sum.
If now the number of dividing points increases indefinitely, and the length max (highest) partial cut partitioning tends to zero, and the value tends to a certain limit does not depend on the method of division and the choice of points on partial segments, then the value is called the area of the curvilinear trapezoid, i.e.
The Formula Newton - Leibniz
If the function is defined and continuous on the interval and is its integral (i.e. ), then
Example. As one of the primitive then
The basic properties of the definite integral
- If integrated on and then