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Find the volume of y=sinx rotating body around y axis. How does it spin? How did you get this formula?
What rotates around the Y axis is a hollow rotator, so it should be a large rotator minus a small rotator. The large rotator is obtained by rotating y=sinx from π/2 to π (that is, x=π-arcsiny) around the y axis, and the small rotator is obtained by rotating y=sinx from 0 to π/2 (that is, x=arcsiny) around the y axis.

The range of arcsiny is [-π/2, π/2]. When x is between π/2 and π, π-x is between 0 and π/2, and y=sinx=sin(π-x), then π-x=siny?

The volume of the Y=sinx rotating body around the y axis is solved as follows:

Extended data

Sine, a mathematical term, in a right triangle, the ratio of the opposite side to the hypotenuse of any acute angle ∠A is called sine of ∠A, which is abbreviated from the English word sine, that is, the opposite side/hypotenuse of Sina = ∠ A.

Generally speaking, in rectangular coordinate system, given the unit circle, for any angle α, the vertex of the angle α coincides with the origin, the starting edge coincides with the non-negative semi-axis of the X axis, and the ending edge intersects with the unit circle at point P(u, v), so the ordinate V of the point P is called the sine function of the angle α, and it is recorded as v=sinα. Usually, we use X to represent the independent variable, that is, X represents the size of the angle and Y represents the function value, so we define the trigonometric function y=sin x at any angle, and its definition domain is all real numbers, with the value range of [- 1, 1].

Sine function correlation formula:

Sum of squares relation

(sinα)^2 +(cosα)^2= 1

Product relationship

Sinα = tanα × cosα (that is, sinα/cosα = tanα)

Cosα = cotα × sinα (that is, cosα/sinα = cotα).

Tanα = sinα × secα (that is, tanα/sinα = secα).

Reciprocal relationship

tanα × cotα = 1

sinα × cscα = 1

cosα × secα = 1

Relationship of quotient

sinα / cosα = tanα = secα / cscα

Sum angle formula

sin ( α β ) = sinα cosβ cosα sinβ

sin(α+β+γ)= sinαcosβcosγ+cosαsinβcosγ+cosαcosβsinγ-sinαsinβsinγ

cos ( α β ) = cosα cosβ? Octagonal β-Octagonal α

tan ( α β ) = ( tanα tanβ ) / ( 1? tanα tanβ)

Double-angle and half-angle formulas

sin ( 2α ) = 2sinα cosα = 2 / ( tanα + cosα)

sin(3α)= 3 sinα-4 sin & amp; sup3(α ) = 4sinα sin ( 60 + α ) sin ( 60 - α)

sin ( α / 2 ) = √( ( 1 - cosα ) / 2)

Derived from Taylor series

Sinks = [East (9)-East (-9)]/(2i)

series expansion

sin x = x - x3 / 3! + x5 / 5! -...(- 1)k- 1 * 2k- 1/(2k- 1)! +...(-∞& lt; x & lt∞ )

derivant

(sinx ) ' = cosx

(cosx)' =-Sinks

References:

Baidu encyclopedia sine function