Double Integrals over General Regions.htm

Double Integrals over General Regions: Examples

1. Find the volume of the solid region bounded by the paraboloid $z=4-x^{2}-2y^{2}$ and the xy plane.

Answer:

By setting z=0, we find that the region the paraboloid takes up on the xy is a ellipse $4=x^{2}+2y^{2}.$ Thus, the region of integration is:

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Thus the volume is given by the following integral:
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Note: $x=2\sin \theta$
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2. Find the volume of the solid region R bounded above by the paraboloid $z=1-x^{2}-y^{2}$ and below by the plane .

Answer:

First find the bounds by equating the z values:

R is the difference between the volume under the paraboloid and the volume above the plane, you have:
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Note: $2y-1=\sin \theta$
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