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Additional resources for Calculus 2c-6, Examples of Space Integrals
Com 47 Calculus 2c-6 Space integrals, spherical coordinates By the reduction theorem in spherical coordinates, 2π 3 (x2 + y 2 + z 2 )− 2 dΩ = 0 A b = 2π 1 1 dr · r π 0 r−3 r2 sin θ dr dθ dϕ B∗ sin θ dθ = 2π[ln r]ba · [− cos θ]π0 = 4π ln b a . 5) Here A is that part of the ball of centrum (0, 0, 0) and radius a, which lies in the upper half space, thus 0 ≤ ϕ ≤ 2π, and Please click the advert B ∗ = B ∗ (ϕ) = (r, θ) 0 ≤ r ≤ a, 0 ≤ θ ≤ π 2 . We have ambitions. Also for you. SimCorp is a global leader in ﬁnancial software.
D Identify the point set. Sketch if necessary the meridian cut. Finally, compute the space integral by reduction in spherical coordinates. √ I 1) It is obvious that A is a conic slice of the ball of centrum (0, 0, 0) and radius 2. 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 − .
5) Here A is that part of the ball of centrum (0, 0, 0) and radius a, which lies in the upper half space, thus 0 ≤ ϕ ≤ 2π, and Please click the advert B ∗ = B ∗ (ϕ) = (r, θ) 0 ≤ r ≤ a, 0 ≤ θ ≤ π 2 . We have ambitions. Also for you. SimCorp is a global leader in ﬁnancial software. At SimCorp, you will be part of a large network of competent and skilled colleagues who all aspire to reach common goals with dedication and team spirit. We invest in our employees to ensure that you can meet your ambitions on a personal as well as on a professional level.