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Centroid, the x-direction centroid distance of the area surrounded by y = ax^n and the y-axis

작성자 Uploader : pleiades 작성일 Upload Date: 2019-03-07변경일 Update Date: 2019-10-22조회수 View : 267

As shown in the figure, the x-direction centroid distance Gx of the area surrounded by y = ax^n and the y-axis can be obtained as follows. 

Gx = My / A

My : first moment of inertia relative to y axis  
A : area

My = ∫ (y1-y)*x dx = ∫ a(x1^n - x^n)*x dx

   = a ∫ x1^n*x - x^(n+1) dx

   = a(x1^n*(1/2)*x^2 - (1/(n+2))*x^(n+2))

Since x = x1,

My = a((1/2)*x1^(n+2) - (1/(n+2))*x1^(n+2))
 
   = a*((1/2)-(1/(n+2)))*x1^(n+2)

   = a*n/(2(n+2))*x1^(n+2)

Firstly, calculating the area to the x-axis :

Ax = ∫ y dx = ∫ a*x^n dx = (a/(n+1))*x^(n+1)

Then, the area from 0 to x1 is :

Ax = (a/(n+1))*x1^(n+1)

Subtracting this from x1*y1 :

A = x1*y1 - Ax = x1*y1 - (a/(n+1))*x1^(n+1)

Since y1 = ax1^n,

A = ax1^(n+1) - (a/(n+1))*x1^(n+1)

 = a*(1-1/(n+1))*x1^(n+1)

 = a*(n/(n+1))*x1^(n+1)

Gx = a*(n/(2(n+2)))*x1^(n+2) / (a*(n/(n+1))*x1^(n+1))

   = 1/(2(n+2)))*x1 / (1/(n+1)))

   = ((n+1)/(2(n+2)))*x1

*** 참고문헌[References] ***
Gx = x1*(n+1)/(2*n+4)
변수명 Variable 변수값 Value 변 수 설 명 Description of the variable


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