Answered: Consider I =E∫x32dx1dx2dx3 , where x12…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider I =_E∫x₃²dx₁dx₂dx₃, where x₁² + x₂² ≤ x₃², 0 ≤ x₃ ≤ 1 ,that is, E is the solid bounded by the cone x₁² + x₂² = x₃², the plane x₃=0, and the plane x₃=1

Some data may be needed (see image below): In the formulas below E, S and l always denote a solid, a surface, and a line, respectively. While n(x) denotes the normal unitary exterior of S in x, and T(x) denotes the unitary tangent of l in x.

Vg(z)
1. S: g(x) = 0, n(x) = +T: E, l: y(t), te [a, b], T(x) = O
r = y(t).
%3D
2. |n(x).k|ds = dr,dr2, n(r).j|ds = dxıdx3, |n(x).i|ds = drzdr3, i = (1,0,0), j = (0, 1,0), k =
(0,0, 1), .
%3D
3. T(1).idl = dr1, T(x).jdl = dr2, T(x).kdl = dr3, donde dl = ||/(t)||dt, .
4. (Gauss) / V.F(r)dx = S F(r).n(1)ds, onde dr =
dz_drzdr3, V = i+ j+k
S=ƏE

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