tasks that lead to the concept of "double integral".
- Let the plane defined flat plate material at each point where the density is known.We need to find a lot of this record.Since this disc has the exact dimensions, that it can be enclosed in a rectangle.The density of the plate can be understood also as follows: at the points of the rectangle, which do not belong to the plate, we assume that the density is zero.Define breaking even on the same number of particles.Thus, the predetermined shape is divided into elementary rectangles.Consider one of these rectangles.We choose any point of the rectangle.Due to the small size of the rectangle, we assume that the density at each point of the rectangle is constant.Then, a rectangular mass of the particles, will be defined as the multiplication of the density at this point in the area of a rectangle.The area is known, multiplying this by the width of the rectangle length.And on the coordinate plane - a change with some steps.Then the weight of the whole record will be the sum weight of the rectangles.If in such a ratio to move to the edge, then we can get the exact ratio.
- We define spatial body, which is limited to the origin and some function.We need to find the volume of said body.As in the previous case, we divide the area into rectangles.We assume that the points that do not belong to the region, the function will be equal to 0. Let us consider one of the rectangular broken.Through the side of the rectangle draw planes that are perpendicular to the axes of abscissa and ordinate.We get a box that is bounded from below with respect to the plane of the Z-axis, and the top of the function, which was defined in the problem statement.Choose a point in the middle of the rectangle.Due to the small size of the rectangle can be assumed that the function within this rectangle has a constant value, then you can calculate the amount of the rectangle.The volume figure will be equal to the sum of volumes of all such rectangles.To get the exact value, you must go to the border.
As can be seen from the objectives, in each instance, we conclude that the various problems lead to the consideration of double sums of the same species.
Properties of the double integral.
pose the problem.Suppose that in a closed area is given a function of two variables, with those given a continuous function.Since the area is limited, it is possible to place it in any rectangle that completely contains the properties of a given point in the area.We divide the rectangle into equal parts.We say that the greatest diameter of breaking the diagonal of the resulting rectangles.Now choose within a single point of the rectangle.If you find the value at this point is to lay down the amount, then such amount will be called integral for a function in a given area.The boundaries of such an integrated amount under the conditions that the diameter of the break should be to 0, and the number of rectangles - to infinity.If such boundary exists and does not depend on the method of breaking the field into rectangles and the choice point, then it is called - a double integral.
geometric content of the double integral: double integral numerals equal to the volume of the body, which was described in the problem 2.
Knowing the double integral (definition), you can set the following properties:
- constant can be taken outside the integral sign.
- integral sum (difference) equal to the sum (difference) integrals.
- of the functions that will be less, which is smaller than the double integral.
- module can be made under the sign of the double integral.
