Let E be the region bounded above by x^(2)+y^(2)+z^(2)=10^(2), within x^(2)+y^(2)=3^(2), below by the xy plane. Find the volume of E.

Question

Let E be the region bounded above by x^(2)+y^(2)+z^(2)=10^(2), within x^(2)+y^(2)=3^(2), below by the xy plane.

Find the volume of E.

Triple Integral

Cylindrical Coordinates

Note: The graph is an example. The scale may not be the same for your particular problem. Round your

answer to one decimal place.

Hint: Convert from rectangular to cylindrical coordinate system.

1. I got (438pi) But it's wrong

Answer 1

Solve using two steps ...

1) find the volume of the cylinder above x²+y²=9 from z=0 to z=sqrt(91) to be 9√91π. Use cyl coord with dV=ρdρdzdΦ and 0≤ρ≤3 ... 0≤z≤√91 ... 0≤Φ≤2π

2) find the volume of the spherical cap above the cylinder. Use cyl coord with dV=ρdρdzdΦ ... 0 ≤ρ≤√100-z²... √91 ≤ z ≤ 10 .... 0≤Φ≤2π ... vol=2.09π

3) add the two to find vol = 87.9π