Use cylindrical coordinates to find the volume of the solid

Use cylindrical coordinates to find the volume of the solid. solid inside the sphere x^2 + y^2 + z^2 = 25 and above the upper nappe of the cone z^2 = x^2 + y^2

Solution

Solution:

In spherical coordinates, the equation of the sphere is

rho²⁼x² +y² +z² =25

rho=5

Furthermore, the sphere and cone intersect when

(x2 + y2) + z2 = (z2) + z2 =25( upper nappe of the cone x² +y² =z²⁾

z=5/sqrt(2)

and, because z = cos( phi), it follows that

=(5/sqrt(2))(1/5) =cos (phi)

phi=pi/4

Consequently, you can use the integration order d dphi d,

where 0 3, 0 phi /4,

and 0 2.

The volume is v=integral v(dv)= integral(o to 2pi)integral(0 to pi/4)integral(0 to 5)(rho² sin phi d dphi d)

=2pi integral(0to pi/4)(rho³/2)₀⁵[sin(phi )dphi)

=(125x2pi)/3(1-sqrt(2)/2)=(125x2pi)/3(2-sqrt(2))/2

=(125xpi)/3(2-sqrt(2)=130.8 (2-sqrt(2)