If one of the three vertices of an equilateral triangle is at the origin, find the area of the triangle

If one of the three vertices of an equilateral triangle is at the origin, find the area of the triangle

∵ is an equilateral triangle ∵ all three angles are 60 degrees ∵ you can make a vertical line through the vertex to get y = root 3 2x, y ^ 2 = 2x

The volume of the body of revolution formed by the ellipse X & # 178 / 4 + Y & # 178 / 9 = 1 rotating around the Y axis is

V=4/3πab²=4/3π*2*9=24π；
This formula can be derived by the method of definite integral, and can also be compared with the volume formula of sphere
The basic idea is to regard the ellipsoid as composed of many small slices. After integration, the volume formula can be obtained

Find the volume of the body of revolution generated by the curve X ^ 2 + (y-b) ^ 2 = a ^ 2 rotating around the X axis

Let the volume of the body of revolution be v. according to the symmetry of the circle x ^ 2 + (y-b) ^ 2 = a ^ 2, we only need to consider the body of revolution of the semicircle, and then multiply it by 2
V=2π\int_ 0 ^ a [(a ^ 2-x)_ 2) ) - (a ^ 2-x under b-radical)_ 2))]^2dx=16πa^3/3.
Comment: int_ 0 ^ A is the definite integral from 0 to A. I hope you can understand the integral expression I wrote

Calculate the revolving body (called revolving ellipse) formed by the figure surrounded by the ellipse x ^ 2 / A ^ 2 x y ^ 2 / b ^ 2 = 1 rotating around the X axis
The calculation is based on the ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1
The volume of a revolving body (called a revolving ellipsoid) formed by revolving the figure around the x-axis

(1)
Let x = x / A, y = Y / b
S = ∫∫ DXDY (where x is from - A to a and Y is from - B to b)
=Ab ∫ DXDY (where x is from - 1 to 1 and Y is from - 1 to 1)
=AB * area of a circle with radius 1
=πab
（2）
Let x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 + Z ^ 2 / C ^ 2 = 1
Let x = x / A, y = Y / B, z = Z / C
V = ∫∫∫ dxdydz (where x from - A to a, y from - B to B, z from - C to c)
=ABC ∫∫∫ dxdydz (where x from - 1 to 1, y from - 1 to 1, z from - 1 to 1)
=ABC * volume of a sphere with radius 1
=(4/3)πabc
Ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1, the volume of the body of revolution rotating around the X and Y axes respectively
They are: (4 / 3) π AB ^ 2, (4 / 3) π Ba ^ 2
Or it can be calculated as follows:
The ellipsoid equation obtained by X & # 178; / A & # 178; + Y & # 178; / B & # 178; = 1 rotating around the x-axis
x^2/a^2+y^2/b^2+z^2/b^2=1
Let x = x / A, y = Y / B, z = Z / b
V = ∫∫∫ dxdydz (where x from - A to a, y from - B to B, z from - B to b)
=AB ^ 2 ∫∫∫ dxdydz (where x from - 1 to 1, y from - 1 to 1, z from - 1 to 1)
=AB ^ 2 * the volume of a sphere with radius 1
=(4/3)πab^2
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