Aerospace Equipment

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

By Mejlbro L.

Show description

Read or Download Calculus 2c-6, Examples of Space Integrals PDF

Best aerospace equipment books

Introduction to Hilbert Space: And the Theory of Spectral Multiplicity

A transparent, readable introductory therapy of Hilbert house. The multiplicity thought of continuing spectra is taken care of, for the first time in English, in complete generality.

Knowledge of Time & Space: An Inquiry into Knowledge, Self & Reality

Within the cosmic dance of time and area, how does wisdom take shape? the connection among intimacy, nice love, and information.

Geology and Habitability of Terrestrial Planets (Space Sciences Series of ISSI) (Space Sciences Series of ISSI)

Given the basic significance of and common curiosity in even if extraterrestrial existence has built or may possibly finally strengthen in our sun method and past, it is important that an exam of planetary habitability is going past basic assumptions resembling, "Where there's water, there's lifestyles.

Additional resources for Calculus 2c-6, Examples of Space Integrals

Example text

D I have here found four variants: 1) Reduction in spherical coordinates. 2) Reduction in semi-polar coordinates. 3) Reduction by the slicing method. 4) Reduction in rectangular coordinate. These methods are here numbered according to their increasing difficulty. The fourth variant is possible, but it is not worth here to produce all the steps involved, because the method cannot be recommended i this particular case. I First variant. Spherical coordinates. The set A is described in spherical coordinates by r ∈ [a, 2a], ϕ ∈ [0, 2π], θ ∈ 0, (r, ϕ, θ) π 2 , hence by the reduction of the space integral, A z dΩ = 2 2 c + x + y2 + z2 π 2 = 2π 0 sin2 θ 2 = 2π = π 2 cos θ sin θ dθ · 4a2 a2 π 2 0 1− · 4a2 a2 c2 c2 + t π 2 2π 0 a 0 2a a c2 2a r cos θ · r2 sin θ dr dθ c2 + r2 r2 · r dr + r2 dϕ [t = r 2 ] t + c 2 − c2 1 · dt c2 + t 2 dt = π t − c2 ln c2 + t 2 4a2 t=a2 = π 2 3a2 − c2 ln 4a2 + c2 a2 + c2 .

C2 −x2 0 |y| ≤ √ c2 −z 2 c J z· d z dΩ = 2 1 z(c2 − z 2 ) dz · · 1 − t2 dt dz 1 − t2 dt = c4 · 0 π , 4 where there are lots of similar variants. 2) First variant. A symmetric argument. The set A is symmetric with respect to the planes y = 0 and x = 0, and the integrand x, resp. y, is an odd function. Hence, x dΩ = 0 and A y dΩ = 0. A Second variant. Spherical coordinates. By insertion, π 2 2π x dΩ = 0 A 0 2π = 0 c 0 cos ϕ dϕ · r sin θ cos ϕ · r 2 sin θ dr dθ π 2 0 sin2 θ dθ · c 0 dϕ r3 dr = [sin ϕ]2π 0 · π c4 · = 0, 4 4 and similarly.

Reduced by the factor 1 − h area(B(z)) = 1 − z h 2 area B = 1 − z h 2 A. com 63 Calculus 2c-6 Volume 2) Using the result from 1) we get by the slicing method, h vol(K) h dΩ = = dx dy 0 K h = 0 1− z h dz = area((B(z)) dz 0 B(z) 2 A dz = Ah − 3 h z 1 1− 3 h = 0 1 hA. 3 3) Let the cone be homogeneously coated (density μ > 0). Then the mass is M = μ vol(K) = 1 μhA. 3 The z-coordinate ζ of the centre of gravity is given by M ·ζ =μ z dΩ, K thus ζ = μ M z dΩ = K h z z 1− h h = 3 0 1 3 1 4 t − t 3 4 = 3h 1 3 μ μhA 2 h 0 1 dz = 3h 0 1 (1 − t)t2 dt = 3h 1 1 − 3 4 = 3h 0 = h 3 hA z · areal(B(z)) dz = 1 0 0 z 1− z h 2 A dz (t2 − t3 ) dt h .

Download PDF sample

Rated 4.40 of 5 – based on 22 votes