Triple Integrals in Cylindrical and Spherical Coordinates.htm

Triple Integrals in Cylindrical and Spherical Coordinates: Examples

1. Fond the volume of the solid region Q cut from the sphere: by the cylinder $r=2\sin \theta$ .

Solution:

Because the bounds on z are:

Let R be the circular projection of the solid onto the r $\theta$ -plane. Then the bounds on r and $\theta$ are:

Thus the integral is:
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2. Find the volume of the solid region Q bounded below by the upper nape of the cone $z^{2}=x^{2}+y^{2}$ and above by the sphere

Answer:

In spherical coordinates, the equation of the sphere is:

Furthermore, the sphere and cone intersect when:
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and, because $z=\rho \cos \phi$ , it follows that:
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Consequently, you can use the integration order . where and 0:
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