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## Triple integrals in spherical coordinates.

Example 7.20   Let be the cone whose equation is . By substitution from  2.6, we have:

i.e., an equation for in cylindrical coordinates is the following:

Example 7.21   See Fig.  16.

• If R is a positive real number, is the equation of a sphere, whose center is at the origin.
• If is a real number, is the equation of a cone, whose vertex is at the origin, and which makes an opening angle of with the positive z-axis.
• If is a real number, is the equation of a half-plane, whose border is the z-axis, and and which makes an angle of with the positive x-axis.

Example 7.22   Compute the volume of the solid cut from the sphere whose equation is by the cone .

The domain of integration is given by the conditions:

The volume is:

Next: Substitution in multiple integrals. Up: Triple integrals. Previous: Triple integrals in cylindrical
Noah Dana-Picard
2001-05-30