Use the shell method to find the volume of y=x^4?

Volume , V , is given by :-

2

∫ π y² dx

1

2

∫ π x^8 dx

1

π [ x^9/9 ]________limits 1 to 2

π [ 2^9/9 - 1/9 ] = [ 511/9 ] π units³

Graph y = x⁴ , y=0 , x=1 and x=2 (rotation axis)

x=1

y=x⁴ gives (x, y) = (1, 1)

x=2

y=x⁴ gives (x, y) = (2, 16)

The bounded volume contains of 2 parts a) from x=0 to x=1 a cylinder with volume =π *1²*1 = π

and b) the volume from x=1 to x=2.

The volume element dV is a cylinder shell with radius = 2 - x = 2 - y^(1/4) and height dy.

dV = 2π (2 - y^(1/4)) dy integrate from y=1 to y=16

V = ∫ 2π ( 2 - y^(1/4)) dy = 2π [2y - 4/5*y^(5/4) ] = 2π(2*16 - 4/5*16^(5/4) - (2*1 - 4/5*1^(5/4) )

V = 2π (32 - 4/5*32 - 2 + 4/5*1) = 2π (30 - 4/5*31) = 2π*26/5 = π*52/5

The total volume: π*52/5 + π = π*57/5 ≈ 35.8 v.u

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