Mathematical Methods for Physics and Engineering : A Comprehensive Guide

6.3 APPLICATIONS OF MULTIPLE INTEGRALS and letN→∞as each of the volumes ∆Vp→0. If the sumStends to a unique limit,I, then this i ...

MULTIPLE INTEGRALS z c dx a x dz dy b y dV=dx dy dz Figure 6.3 The tetrahedron bounded by the coordinate surfaces and the planex ...

6.3 APPLICATIONS OF MULTIPLE INTEGRALS the total volume of the tetrahedron is given by V= ∫a 0 dx ∫b−bx/a 0 dy ∫c( 1 −y/b−x/a) 0 ...

MULTIPLE INTEGRALS z 0 2 y dV=dx dy dz z=2y z=x^2 +y^2 x Figure 6.4 The region bounded by the paraboloidz=x^2 +y^2 and the plane ...

6.3 APPLICATIONS OF MULTIPLE INTEGRALS The coordinates of the centre of mass of a solid or laminar body may also be written as m ...

MULTIPLE INTEGRALS z x y a a a √ a^2 −z^2 dz Figure 6.5 The solid hemisphere bounded by the surfacesx^2 +y^2 +z^2 =a^2 and thexy ...

6.3 APPLICATIONS OF MULTIPLE INTEGRALS y y ds ̄y x Figure 6.7 A curve in thexy-plane, which may be rotated about thex-axis to fo ...

MULTIPLE INTEGRALS a θ d C Figure 6.8 Suspending a semicircular lamina from one of its corners. 6.3.4 Moments of inertia For pro ...

6.4 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS y x b dx a dM=σb dx Figure 6.9 A uniform rectangular lamina of massMwith sidesaand ...

MULTIPLE INTEGRALS y x u=constant v=constant N M L K R C Figure 6.10 A region of integrationRoverlaid with a grid formed by the ...

6.4 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS is constant along the line elementKL, the latter has components (∂x/∂u)duand (∂y/∂ ...

MULTIPLE INTEGRALS
Evaluate the double integral I= ∫∫ R ( a+ √ x^2 +y^2 ) dx dy, whereRis the region bounded by the circlex^2 + ...

6.4 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS a a −a −a y x Figure 6.11 The regions used to illustrate the convergence propertie ...

MULTIPLE INTEGRALS z x y C R T S P Q u=c 1 v=c 2 w=c 3 Figure 6.12 A three-dimensional region of integrationR, showing an el- em ...

6.4 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS ware constant, and soPQhas components (∂x/∂u)du,(∂y/∂u)duand (∂z/∂u)du in the dire ...

MULTIPLE INTEGRALS which agrees with the result given in chapter 10. If we place the sphere with its centre at the origin of anx ...

6.5 EXERCISES and similarly forJyzandJxz. On taking the determinant of (6.14), we therefore obtain Jxz=JxyJyz or, in the usual n ...

MULTIPLE INTEGRALS 6.6 The function Ψ(r)=A ( 2 − Zr a ) e−Zr/^2 a gives the form of the quantum-mechanical wavefunction represen ...

6.5 EXERCISES This is an example of the general result for planar bodies that the moment of inertia of the body about an axis pe ...

MULTIPLE INTEGRALS over the ellipsoidal region x^2 a^2 + y^2 b^2 + z^2 c^2 ≤ 1. 6.18 Sketch the domain of integration for the in ...