Double Integrals in Polar Coordinates.htm

Double Integrals in Polar Coordinates: Examples

1. Let R be the annular region lying between the two circles $x^{2}+y^{2}=1$ and $x^{2}+y^{2}=5$ .

Answer:

The polar boundaries are and Also, and $y=r\sin \theta :$
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2. Use change of variables to find the volume of the solid region bounded above by the hemisphere and below by the circular region R given by $x^{2}+y^{2}\leq 4.$

Answer:

The region $x^{2}+y^{2}\leq 4$ is simply a circle in the xy plane with a r=2. Thus the bounds would be $0\leq r\leq 2$ and $0\leq y\leq 2\pi .$ In addition, the hemisphere which forms the top of the cylinder can be changed into polar coordinates:

Thus, the volume of the sphere is given by the following integral:

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