By Leif Mejlbro

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**Additional resources for Real Functions of Several Variables Examples of Space Integrals Calculus 2c-6**

**Example text**

That part A1 of A, which is given by ∈ [0, a[, is described in semi-polar coordinates by {( , ϕ, z) | ∈ [0, a[, ϕ ∈ [0, 2π], That part A2 of A, which is given by {( , ϕ, z) | a2 − 2 ≤z≤ 4a2 − 2 }. ∈ [a, 2a], is described in semi-polar coordinates by ∈ [a, 2a], ϕ ∈ [0, 2π], 0 ≤ z ≤ 4a2 − 2 }. 5 Figure 32: The meridian cut for a = 1 with the line x = a = 1. Then by reduction in semi-polar coordinates, A z dΩ = c2 + x2 + y 2 + z 2 √ 2 2π a √ = 0 A1 4a − 0 a2 − 2π z dΩ + c2 +x2 +y 2 +z 2 2 2 z √ 2a 4a2 − 2 + 0 a = 2π 0 a =π 0 1 1 ln c2 + 2 0 2 +z dz c2 + 2 +z 2 √ 2 z= √ dz 2 +z 2 2 d 2a d + 2π a2 − ln c2 +4a2 −ln c2 +a2 dϕ z c2 + 4a2 − d A2 z dΩ c2 +x2 +y 2 +z 2 a 2 2a d +π a dϕ √ 1 ln c2 + 2 2 +z 2 4a2 − d z=0 ln c2 +4a2 −ln c2 + 2 d , thus A z dΩ c2 + x2 + y 2 + z 2 = π ln c2 +4a2 −ln c2 +a2 π − 2 4a2 · a2 1 + π ln c2 +4a2 · 4a2 −a2 , 2 2 ln c2 + t dt (ved t = 2 ) a1 π π π · 4a2 ln c2 +4a2 − a2 ln c2 +a2 − c2 + t ln c2 + t −t 2 2 2 π π 2 π c +4a2 ln c2 +4a2 = · 4a2 ln c2 +4a2 − a2 ln c2 +a2 − 2 2 2 π π π 2 2 c +a ln c2 +a2 − · a2 + · 4a2 + 2 2 2 c2 + 4a2 π 2 2 = · 3a − c ln .

D Sketch Ω, or at least the projection D of Ω onto the (x, y)-plane.

And One shall ﬁrst from the given description of the domain of integration ﬁnd α, β and B(ϕ). 1) The space integral x2 + y 2 dΩ, where the domain of integration A is given by A x2 + y 2 ≤ z ≤ 1. 2) The space integral A ln(1 + x2 + y 2 ) dΩ, where the domain of integration A is given by 1 2 (x + x2 ) ≤ z ≤ 2. 2 3) The space integral x2 + y 2 ≤ 1 A (x + y 2 )z dΩ, where the domain of integration A is given by x2 + y 2 ≤ z ≤ and 4) The space integral A 2 − x2 − y 2 . (x2 + y 2 ) dΩ, where the domain of integration A is given by x2 + y 2 ≤ z ≤ h.