High Number Solving Why does the double integral use the function parity to appear even odd zeros?

Odd functions. The number of points will be 0. Even if it is not an odd function, the integral may still be 0. When the integral region is symmetric with respect to the x-axis, if the integrand is an odd function with respect to y, then the integral value is 0; If the integrand is an even function about y.

The integral value is twice that of this part of the symmetry region.

When the integrand is greater than zero, the double integral.

is the volume of the cylinder. When the integrand is less than zero, the double integral is the negative value of the cylinder volume.

Some special formulas for the volume of the top cylinder of the curved top surrounded by the surface represented by the integrand f(x,y) and the bottom surface of d.

It is known that it can be calculated in terms of the geometric meaning of the double integral.

The same principle as the definite integral.

On [-a,a], if f(x) is an odd function, f(-x)=-f(x), a,a) f(x) dx, let x=-u.

a，-a) f(-u)*(du)

-a，a) f(-u) du

-a，a) -f(u) du

(a,a) f(x) dx, move to (-a,a) f(x) dx=0.

In the same way, (-a,a) f(x) dx = 2 (0,a) f(x) dx if f(x) is an even function.

As for the double integral, if d is symmetrical with respect to both the x-axis and the y-axis.

Moreover, if the integrand is an odd function about x or y, the result is 0 as well.

For example, d is x 2 + y 2 = 1

Then x, x 3, xy, xy 3, y 5, x 3y 3, and so on are all 0.

Don't think that xy and x 3y 3 are even functions, parity is valid for a single independent variable.

Y is used as a constant when calculating x, so when the integral result of x is 0, there is no need to integrate y.

f(x)=xy, which treats y as a constant, so xy is an odd function about x.

g(x)=cosxsiny, which is considered to be a constant.

So cosxsiny is an even function with respect to x.

g(y) = cosxsiny, which treats cosx as a constant.

So cosxsiny is an odd function with respect to y.

1. If the integral region is symmetrical with respect to the x-axis.

The integrand is an odd function with respect to y, which is equal to 0; The even function of the integrand with respect to y is equal to 2 times.

2. If the integration region is symmetrical with respect to the y-axis.

The integrand is the odd function of the silver potato with respect to x, which is equal to 0; The even function of the integrand with respect to x is equal to 2 times.

3. If the integration region is symmetrical with respect to the x,y axis.

The integrand is about the odd function of thinking about x,y , which is equal to 0; The even function of the integrand with respect to x,y is equal to 2 times.

Double integral meaning.

When the integrand is greater than zero, the double integral is the volume of the cylinder.

When the integrand is less than zero, the double integral is the negative value of the cylinder volume.

Geometric significance. In a spatial Cartesian coordinate system, the double integral is the algebraic sum of the volume of the cylinder over each part region, which is positive above the xoy plane and negative below the xoy plane. The formulas for the volume of some special surface of the integrand f(x,y) and the volume of the top cylinder surrounded by the bottom of d are known, and can be calculated without the geometric meaning of the double integral.

For example, double integral:

Thereinto. <>

It represents a curved top cylinder with the upper hemisphere as the top and the circle with radius a as the bottom, and this double integral is the volume of the hemisphere.

The integration of an even function on a symmetric interval is equal to 2 times its integration on half of the entire interval. The y-equal old cosx is an even function, and its integration on an arbitrary symmetry interval (negative a,a) (a greater than 0) is equal to 2 times the integral stool on (0,a).

The integral of an odd function will be 0. Even if it is not an odd function, the integral may still be 0. When the integral region friend shouts symmetry about the x-axis, if the integrand is an odd function with respect to y, then the integral value is 0; If the integrand is an even function with respect to y, the integral value is twice as much as "this part of the symmetry region".

When the integrand is greater than zero, the double integral is the volume of the cylinder. When the integrand is less than zero, the double integral is the negative value of the column and filial piety volume.

The volume formulas of some special integrands f(x,y) representing the surface and the bottom surface of d are known and can be calculated by the geometric meaning of the double integral.

This is the property of the integral, no matter how many integrals there are, as long as the slippery hidden product function is an odd function, and the integral interval is symmetrical with respect to the origin of the deficit, the result is 0.

The integrand is an even function, and the integral interval is called to the letter about the origin, and the integral = 0 of 2 times to the upper limit of the integral = 0 of 2 times to the upper limit of the integral = 0 of 2 times to the upper limit.

The calculation of the double integral is the same as in the above form.

The linear nature of the integral.

Property 1, (integral additivity) The double integral of the sum (difference) of the function is equal to the sum (difference) of the double integral of each function.

Property 2, (multiplication of integral satisfaction) The constant coefficient factor of the integrand can be mentioned outside the integral sign, i.e. (k is a constant).

Property 3, let m and m be the maximum and minimum values of the function f(x,y) on the bounded closed region d, respectively, and be the area of the region d.

Odd zero and even multiple principle: the fixed product dilemma difference refers to the algebraic sum of the image area of the function, and the odd function image is symmetrical according to the origin, so when the absolute value of the upper and lower limits of the integral is equal, the two sides of the graph area cancel each other out, and the value is zero. The even function is symmetric according to the y-axis, and the definite integral is doubled.

In the calculation of definite integrals, if the integral interval is symmetrical with respect to the origin and the continuous function is not non-odd or even over the defined interval. then you can flexibly use even times odd zeros.

Riemann Points. The official name for a definite integral is the Riemann integral.

In Riemann's own words, the image of the function in the Cartesian coordinate system is divided into an infinite number of rectangles with a straight line parallel to the y-axis, and then the rectangles on a certain interval [a, b] are added up, and the area of the image of this function number in the interval [a, b] is obtained. In fact, the upper and lower bounds of the definite integral are the two endpoints a and b of the interval.