By Mejlbro L.

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**Example text**

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.

We invest in our employees to ensure that you can meet your ambitions on a personal as well as on a professional level. SimCorp employs the best qualiﬁed people within economics, ﬁnance and IT, and the majority of our colleagues have a university or business degree within these ﬁelds. Ambitious? com 48 Calculus 2c-6 Space integrals, spherical coordinates By the reduction theorem in spherical coordinates, 2π (x2 z + z 3 ) dΩ = (r2 sin2 θ cos2 ϕ r cos θ+r 3 cos3 θ)r2 sin θ dr dθ dϕ B∗ 0 A 2π a = 0 = = = = = 0 a r6 6 a6 6 a6 5 π 2 r5 π 2 0 0 π 2 2π 0 0 2π 0 dr dϕ 0 2π 0 (cos2 ϕ sin2 θ cos θ+cos3 θ) sin θ dθ {cos2 ϕ(cos θ−cos3 θ)+cos3 θ} sin θ dθ {cos2 ϕ cos θ+sin2 ϕ cos3 θ} sin θ dθ 1 1 − cos ϕ · cos2 θ − sin2 ϕ · cos4 θ 2 4 2 a6 a6 2π {2 cos2 ϕ + sin2 ϕ}dϕ = 24 0 24 a6 3 a6 π · · 2π = .

2) The space integral is here computed in four variants. com 59 Calculus 2c-6 Space integrals, spherical coordinates First variant. Direct insertion: 2π (x2 + z 2 ) dΩ = = = 2π a5 5 π 2π a5 5 − cos θ + 0 r2 sin2 θ cos2 ϕ + r2 cos2 θ r2 sin θ dr dθ dϕ 0 (1 − cos2 θ) cos2 ϕ + cos2 θ sin θ π 3 0 a π 3 0 A π 1 cos3 θ 3 π π 3 cos2 ϕ + − dϕ π 1 cos3 θ 3 dϕ π 3 1 1 1 a5 2π 1 1 + 1− + − cos2 ϕ + dϕ 5 0 3 2 24 3 24 3 3 3 3 9 a5 a5 9 − · 2π = π − + = π+ 5 2 24 24 5 2 8 4 3π 5 3π 5 1 1 3π 5 5 a + ·a · = a .

### Calculus 2c-6, Examples of Space Integrals by Mejlbro L.

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