A volcano fills the volume between the graphs z=0 and z=1 / (x^2+y^2) ^10 and outside the cylinder x+y=1. find the volume.

For this case, we use the cylindrical coordinates:

x² + y² = r²

dV = r dz dr dθ

The limits are:

z = 0 to z = 1 / (r²) ^10 = 1/r^20

r = 1 to ∞

θ = 0 to 2π

Integrating over the limits:

V = ∫ [0 to 2π] ∫ [1 to ∞] ∫ [0 to 1/r^20] r dz dr dθ

V = ∫ [0 to 2π] ∫ [1 to ∞] rz | [z = 0 to 1/r^20] dr dθ

V = ∫ [0 to 2π] ∫ [1 to ∞] 1/r^19 dr dθ

V = ∫ [0 to 2π] - 1 / (18r^18) |[1 to ∞] dθ

V = ∫ [0 to 2π] 1/18 dθ

V = θ/18 |[0 to 2π]

V = π/9