. Differential and integral calculus, an introductory course for colleges and engineering schools. integrations. To this end we take, as elements of volume AV, the smallparallelopipeds made by planes parallel to the coordinate the distances apart of these planes be Azi, Az2, . . , AyhAz/2, • • • , , Ax2, .... Then the elements of volume willhave the dimensions Az, Ay, Ax, and the products will be of theform/(:c, y, z)AzAy Ax. Since the limit of the sum of productsis independent of the manner of subdividing V, we may write JJJ ^xVi **dV ~ limX^> V> *>Az Ay Ax v v Conside
. Differential and integral calculus, an introductory course for colleges and engineering schools. integrations. To this end we take, as elements of volume AV, the smallparallelopipeds made by planes parallel to the coordinate the distances apart of these planes be Azi, Az2, . . , AyhAz/2, • • • , , Ax2, .... Then the elements of volume willhave the dimensions Az, Ay, Ax, and the products will be of theform/(:c, y, z)AzAy Ax. Since the limit of the sum of productsis independent of the manner of subdividing V, we may write JJJ ^xVi **dV ~ limX^> V> *>Az Ay Ax v v Consider now that slice of V included between the planes x = xkand x = xk + Axk. The section of V made by x = xk is shown inthe figure and is denoted by R. The products belonging to thisslice are X/(^> V> *)Ai2 ^ ^Xk = kzk^Kxk, y, z) Az Ay, where xk and Axk are it is geometrically evidentthat lim^/fo y, z) Az Ay Ax = lim2) Axk Tlim ^f(xk) y, z) Az Ay~\. By Art. 214 we know that thelimit within the brackets is thedouble integral of f(xk, y, z) (xk a constant) extended over R. 334 INTEGRAL CALCULUS §218 (or over the projection of R in the yz-plane). We may writethen \un£f(xk, y, z) Az Ay -ffffa, y, z) dzdy R R y=8(xk) z=\p(xk,y) J J ffrk, y, z) dz dy = F{xk). V=y(xk) z=4>(xk,y) Jr>z=ip(xk,y)f(xk, y, z) dz extends the summation alongz=(xk,y) the line CD from z = 4>(xk, y) at C to z = \p(xk, y) at D. Thesecond integration extends the summation over the whole of R,from y = y(xk) at M or Mi to y = 5(xk) at N or Ni. We havenow limV/(a;, y, z)AzAy Ax = \im^F(x)Ax = f F(x)dx. \r \r Ux=a Hence finally, (D) lim^fix, y, z) Az Ay Ax = j J J f(x, y, z) dz dy dxv v x=b y=8(x) z=>p(x,y) = f f f f(x,y,z)dzdydx. Q. E. D. x=a y=y(x) z=(x,y) By cutting V into slices by planes parallel to the rcz-plane or thexy-pl&ne, it can be shown that it is quite immaterial in whichorder the three successive integrations are performed, care beingtaken in each case to employ prope
Size: 1624px × 1539px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
