calculus

Find the volume of the solid obtained by rotating the region bounded by the curves
y = x^8, y = 1 about
the line y = 5.

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  1. intersection of y =1 and y = x^8 is (1,1) and (-1,1)
    because of the nice symmetry we could just go from 0 to 1 and double.

    V = 2π∫ ( (5-x^8)^2 - (5-1)^2 ) dx from 0 to 1
    = 2π ∫ (25 - 10x^8 + x^16 - 16) dx from 0 to 1
    = 2π [9x - (10/9)x^9 + (1/17)x^17 ] from 0 to 1
    = 2π(9 - 10/9 + 1/17 - 0)
    = 2432π/153

    check my arithmetic

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