Question

Situación: Describa el método de establecer y evaluar para una integral triple. Ofrezca un ejemplo utilizando una de las siguientes opciones: (a) integral triple iterada, (b) uso de coordenadas esféricas, (c) uso de coordenadas cilíndricas. Translation: Situation: Describe the method of establishing and evaluating for a triple integral. Give an example using one of the following options: (a) triple iterated integral, (b) use of spherical coordinates, (c) use of cylindrical coordinates. If you can write it clearly it would be of great help, I need to hand it on Spanish, I'm going to translate it. Thanks!

Answer 1

Calculation of triple integral over any bounded region in Re: If di Esr be a bounded function defined on a closed and regular domain E bounded and above and below by the Surfaces = g(ny) and Z=h(ny), [9 (4,4) > henry) and on the sides by a cylindrical surface, and if I scris fing, 2) dz exists for each fixed point (ny) ED where D is the projection of E on the ny-plane then the iterated integral lots f (x, y, z) dz) dudy also exists and SSS fm, 1, 2) andy de heny) hy, Se les d (1.9, 2) dz) andy him) when the condition of reduction of double integral to the iterated integrals are satisfied che home is fwy, 2) de dy dz - Noo (44) dudy dus (w) dy (play) (m) 4.9. 23 dz] 1 7 ,0) where a line parallel to y. akis cuts the bounding curve of at two points whose y coordinates are 4, (n). 4₂ (2) ( " 4, owlso that 4, (m), Y. (n) becomes the limits of integration wort.y. Thus we finally home Se for 19, 2) du dy dz = l de un des 2 (1) gcny) tex, y, z) dz is the domain a) Example: Evaluate SSS ny dudy dit where bounded by Z=ny, n+y=1, 2=0 (7,0)

solution: SSS ny dn dy dz Side for yrdy = 1 / ² (1-xy dx 1 S (x - 34° + 3x - ns) du at [ 1 - 2 + 1 / 6 7 = 1.80 b) Evaluate die de dy dz where is the region bounded VI-n2g²22 at by the interior of the sphere unty, 42²-1 ) solution: he use the transformation ner sind caso, y=rsine sing, Zareoso which maps the region E to E': borst, orest, OXO5RT Jacobian of the transformation is J = (492) = r²sino so except when r=0, 0-0, 0 = 1 alrood) le, when H2o, y = 0, Z=0. Origin Eis transformed to pole in E, E and E has one to one correspondence. du dy dz der dodo Viore There fore Ile dn dy dz. I. der S" sinodes " de - un sol der - T? c) Evaluate Ie / 1-x2yh audycz where Eis region bounded by the cylinder uity? I, 70, Z=1 solution: he use the transformation Maraust, garsing, zzz

which maps the region Eto E'; osred, oco LRT and of ZL. Jacobian of the transformation is = 2 Chigit) - rzo a(r,0,2) ex cept when no ie. when no, y=0, z=0. Then origin in E is transformed to pole in E, E and El has one to one correspondence alt 2 there fore Ile du dj dz = S/E 131 Var? V2 2 [-V-r] 2 T