please help!!!!!! Use cylindrical coordinates to compute the volume of the solid
ID: 2861048 • Letter: P
Question
please help!!!!!!
Use cylindrical coordinates to compute the volume of the solid E R^3 bounded below by the xy-plane and contained inside the sphere x^2 + y^2 + z^2 = 9 but outside the cylinder x^2 + y^2 = 1. Use spherical coordinates to compute integral integral integral_E (x, y, z)dV in each of the following cases: where f(x, y, z) = z Squareroot x^2 + y^2 + z^2 and E = {(x, y, z) epsilon R^3 | 0 lessthanorequalto x lessthanorequalto Squareroot 9 - y^2, 0 lessthanorequalto y lessthanorequalto 3, 0 lessthanorequalto 2 lessthanorequalto Squareroot 9 - (x^2 + y^2)}; where f(x, y, z) = e^(x^2 + y^2 + z^2)^3/2 and E R^3 is the solid bounded below by the cone z = Squareroot x^2 + y^2 and above by the sphere x^2 + y^2 + z^2 = 1.Explanation / Answer
1)x2+y2+z2=9, xy plane =>z =0
z=[9 -(x2+y2)]
in cylindrical coordinates
x=rcos y=rsin
x2+y2=r2
z=[9 -(r2)] , z=0
[9 -(r2)] =0
r =3
x2+y2=1
r2=1
r =1
0<=<=2,1<=r<=3 ,0<=z<=[9 -r2]
dv =r dz dr d
volume of solid E =[0 to 2][1 to 3][0 to [9 -r2]] r dz dr d
volume of solid E =[0 to 2][1 to 3][0 to [9 -r2]] rz dr d
volume of solid E =[0 to 2][1 to 3] r([9 -r2] -0) dr d
volume of solid E =[0 to 2][1 to 3] ((-1/3)[9 -r2]3/2) d
volume of solid E =[0 to 2] ((-1/3)[9 -32]3/2) - ((-1/3)[9 -12]3/2) d
volume of solid E =[0 to 2] 0 +((1/3)83/2) d
volume of solid E =[0 to 2] ((1/3)83/2)
volume of solid E =(1/3)83/2[2-0]
volume of solid E =(2/3)83/2
volume of solid E =47.39 cubic units
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