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Find volume of solid of revolution of the curve y^2 = (x) (4-x)^2 about y axis.?
If I have to find the volume of solid of revolution about x axis, then it is easy because we know the expression of y in terms of x. But how to find the volume of the solid of revolution about y axis?
1 Answer
- cidyahLv 74 years ago
y = sqrt( x(4-x)^2 )
y = sqrt(16x+x^3-8x^2)
Using shell method:
radius = x
height = sqrt(x) sqrt ((4-x)^2 )
The limits are 0 to 4
Volume = ∫ x sqrt x) sqrt(4-x)^2 ) dx
∫ x sqrt (x) sqrt(4-x)^2 ) dx
Let u= sqrt(x)
u^2 = x
2u du = dx
x= u^2 ; sqrt(x) = u; sqrt((4-x)^2)= 4-x = 4-u^2 ; dx= 2u du
∫ x sqrt (x(4-x)^2) dx = 2∫u^4 (4-u^2) du
= 8 ∫u^4 du - 2 ∫u^6 du
= (8/5) u^5 -(2/7) u^7
replace u by sqrt(x) or x^(1/2)
= (8/5) x^(5/2) - (2/7) x^(7/2)
F(x) = (8/5) x^(5/2) - (2/7) x^(7/2)
F(4) = 512 /35
F(0) = 0
F(4)-F(0) = 512 /35