The proof of 〔∫ (a, b) f (x) DX 〕 & # 178 in double integral; ＝∫(a,b)f(x)dx∫(a,b)f(y)dy

The proof of 〔∫ (a, b) f (x) DX 〕 & # 178 in double integral; ＝∫(a,b)f(x)dx∫(a,b)f(y)dy

As shown in the picture, if not clear, please ask. Please evaluate in time

Let f (x) be continuous on [a, b], and f (b) = a, f (a) = B. It is proved that ∫ (upper B, lower a) f (x) f '(x) DX = 1 / 2 (A & # 178; - B & # 178;)

Integral = ∫ f (x) DF (x) = [f (x)] ^ 2 / 2 = [f (b)] ^ 2 / 2 - [f (a)] ^ 2 / 2 = (a ^ 2-B ^ 2) / 2

It is proved by double integral that: [ʃ (a to b) f (x) DX] ²

On the square [a, b] * [a, b], there is [f (x) -- f (y)] ^ 2 > = 0
The double integral of f ^ 2 (x) + f ^ 2 (y) - 2F (x) * f (y) > = 0
It is easy to calculate that the double integrals of f ^ 2 (x) and f ^ 2 (y) are (B -- a) * integrals (from a to b) f ^ 2 (x) DX
The double integral of F (x) * f (y) is equal to (integral (from a to b) f (x) DX) ^ 2

If ∫ (1 → x) f (T ^ 2) DT = x ^ 3, then ∫ (0 → 1) f (x) DX=
If we know ∫ (1 → x) f (T ^ 2) DT = x ^ 3, then ∫ (0 → 1) f (x) DX = 3 / 2

∫ f (T ^ 2) DT = x ^ 3 = = > F (x ^ 2) = 3x ^ 2
∴f(x)=3x
So ∫ f (x) DX = 3 ∫ xdx
=3*(1/2)
=3/2.

Make a vertical line from a vertex of a triangle to its opposite side. The line between the vertex and the perpendicular foot is called the line of the triangle______ ．

Make a vertical line from a vertex of a triangle to its opposite side. The line between the vertex and the perpendicular foot is called the height of the triangle

From one () to its () make a line (). The line () between () and () is called the height of the triangle. This line () is called the bottom of the triangle
I've been struggling with this problem for a long time,

Make a vertical line from one (vertex) of a triangle to its (opposite side). The distance between (vertex) and (perpendicular foot) is called the height of the triangle, and this (edge) is called the bottom of the triangle

What triangle is a triangle with two equal sides

Isosceles triangle

The sum of the lengths of the three sides of a triangle is called a triangle?

The sum of the three sides of a triangle is called the perimeter of the triangle

Given several elements of a triangle, the process of finding other elements is called ()

Given several elements of a triangle, the process of finding other elements is called solving a triangle

The basic element of a triangle: the line segments that make up the triangle are called ()

The basic element of a triangle: the segments that make up the triangle are called (the sides of the triangle)