Mathematics
KandisMakanani
2016-04-10 05:45:36
Use polar coordinates to find the volume of the given solid. bounded by the paraboloid z = 5 + 2x2 + 2y2 and the plane z = 11 in the first octant.
ANSWERS
ShenaMonnier
2016-04-10 08:20:09

Using cylindrical coordinates, [latex]displaystyleint_{x=0}^{x=sqrt3}int_{y=0}^{y=sqrt{3-x^2}}int_{z=5+2x^2+2y^2}^{z=11}mathrm dz,mathrm dy,mathrm dx[/latex] [latex]displaystyle=int_{ heta=0}^{ heta=pi/2}int_{r=0}^{r=sqrt3}int_{z=5+2r^2}^{z=11}r,mathrm dz,mathrm dr,mathrm d heta[/latex] [latex]=displaystylefracpi2int_{r=0}^{r=sqrt3}r(11-(5+2r^2)),mathrm dr[/latex] [latex]=displaystylepiint_{r=0}^{r=sqrt3}(3r-r^3),mathrm dr[/latex] [latex]=dfrac{9pi}4[/latex]

ADD ANSWER