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coldgirl [10]
3 years ago
6

A solid is bounded below by the cone, z=x2+y2, and bounded above by the sphere of radius 2 centered at the origin. Find integral

s that compute its volume using Cartesian and cylindrical coordinates. For your answers use θ= theta.
Mathematics
1 answer:
ivolga24 [154]3 years ago
4 0

The cone z=\sqrt{x^2+y^2} and the sphere z=\sqrt{4-x^2-y^2} intersect in a circle of radius \sqrt 2 in the plane z=\sqrt2:

\sqrt{x^2+y^2}=\sqrt{4-x^2-y^2}\implies 2x^2+2y^2=4\implies x^2+y^2=2

\implies z=\sqrt{x^2+y^2}=\sqrt2

In Cartesian coordinates, the volume is then given by the integral

\displaystyle\int_{-\sqrt2}^{\sqrt2}\int_{-\sqrt{2-x^2}}^{\sqrt{2-x^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{4-x^2-y^2}}\mathrm dz\,\mathrm dy\,\mathrm dx

In cylindrical coordinates, the integral is

\displaystyle\int_0^{2\pi}\int_0^{\sqrt2}\int_r^{\sqrt{4-r^2}}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta

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