Question

Section 2.3 Consider the solid S bounded by the surface z = 9 - 22 - y2 and the plane 2 = 0. A) Express the volume of S as a double integral(s). B). Determine the volume of S by calculating the integral(s) you found in A). Why does your answer make sense geometrically?

Answer 1

Given that S is the surface bounded by = √ 9-se ² ye and Z= 0 i) We can find the region con notion of me szary setting zo en which gives 19-x?-yz = 0 Solving for y clay moving the square root to the left hand side, squaring both sides, etc.) gives 9-2² g ² = 0 9 -1 = y ut Ja -- > y=+ 9- x The "-" gives the lower limit and "t" gives the upper limit for the outer limits we can see that. -3<x<3.. I we ding this add together, we have J V 9 x y z dy dx - U (i) As we see that the region of integration is a circle So we can convert the xy-coordinate system into polas Coordinates. Put x=rCoso, y=rsino dadya rdr da - Jacobean Also osr <3 , so SOT

V= L / Ware endado v=1 T / 98² & dodo 0=0 do put 9-k² = t -2rde = dt = rea=4jt limits ) d=07 t=9 8-35 t-o ST to Ve T t al dt do 020 t=9 Ž 9 . Iyet's dt do 0=0 to el pr t 319 do 20=0 34 Ito 2J o= z J 0=0 El 27 O=0 27 x 22 - 27T