# Mejlbro L.'s Calculus 2b, Real Functions in Several Variables PDF

By Mejlbro L.

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Com 54 Real Functions in Several Variables The space integral When we use semi-polar coordinates. the domain A is represented by the parametric domain A˜ = {( , ϕ, z) | 0 ≤ ≤ a, 0 ≤ ϕ ≤ π, 0 ≤ z ≤ h}. Then we get at least two possibilities for the reduction. I 1. For ﬁxed ϕ the domain A is cut into B(ϕ) = [0, a] × [0, h]. In this case we get the following reduction where ϕ is kept in the outer integral, π I 0 B(ϕ) π = 0 − = 2 z· = cos2 ϕ · h cos2 ϕ sin ϕ dϕ · π 1 cos3 ϕ 3 0 1 2 z 2 · 0 h 0 sin ϕ · d a z dz · · 1 5 0 5 4 a = 0 dϕ d 2 1 2 1 5 1 2 5 · h · a = h a .

Com 54 Real Functions in Several Variables The space integral When we use semi-polar coordinates. the domain A is represented by the parametric domain A˜ = {( , ϕ, z) | 0 ≤ ≤ a, 0 ≤ ϕ ≤ π, 0 ≤ z ≤ h}. Then we get at least two possibilities for the reduction. I 1. For ﬁxed ϕ the domain A is cut into B(ϕ) = [0, a] × [0, h]. In this case we get the following reduction where ϕ is kept in the outer integral, π I 0 B(ϕ) π = 0 − = 2 z· = cos2 ϕ · h cos2 ϕ sin ϕ dϕ · π 1 cos3 ϕ 3 0 1 2 z 2 · 0 h 0 sin ϕ · d a z dz · · 1 5 0 5 4 a = 0 dϕ d 2 1 2 1 5 1 2 5 · h · a = h a .

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