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Com 54 Calculus 2c-6 Space integrals, spherical coordinates The plane at height z = [0, a[ intersects A in an annulus B(z), which is described in polar coordinates by {( , ϕ) | ϕ ∈ [0, 2π], a2 − z 2 ≤ ≤ 4a2 − z 2 }. The plane at height z ∈ [a, 2a] intersects A in a disc B(z), which is described in polar coordinates by {( , ϕ) | ϕ ∈ [0, 2π], 0 ≤ ≤ 4a2 − z 2 }. Please click the advert it’s an interesting world Where it’s Student and Graduate opportunities in IT, Internet & Engineering Cheltenham | £competitive + benefits Part of the UK’s intelligence services, our role is to counter threats that compromise national and global security.

These can be reduced dramatically thanks to our systems for on-line condition monitoring and automatic lubrication. We help make it more economical to create cleaner, cheaper energy out of thin air. By sharing our experience, expertise, and creativity, industries can boost performance beyond expectations. Therefore we need the best employees who can meet this challenge! The Power of Knowledge Engineering Plug into The Power of Knowledge Engineering. com 43 Calculus 2c-6 Space integrals, semi-polar coordinates π π The meridian cut does not depend on ϕ ∈ − , .

Thus 0 ≤ ϕ ≤ 2π, and the meridian cut √ π 0 ≤ r ≤ 2, 0 ≤ θ ≤ B ∗ = B ∗ (ϕ) = (r, θ) 4 does not depend on ϕ Then by the reduction theorem in spherical coordinates, 2π x2 + y 2 dΩ = A 0 2 N ∗ (ϕ) √ 2 4 = 2π r 4 = π π 1 − 4 2 0 r sin θ · r sin θ dr dθ π 4 0 √ 2 dϕ = 2π 0 4 1 1 − cos 2θ 1 dθ = 2π · · θ − sin 2θ 2 4 2 2 π π2 − . 1. 2. 2) The set A is the unit ball, so ϕ ∈ [0, 2π], and B ∗ = B ∗ (ϕ) is the unit half circle in the right half plane which does not depend on ϕ, B ∗ = B ∗ (ϕ) = {(r, θ) | 0 ≤ r ≤ 1, 0 ≤ θ ≤ π}.

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