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.

Anwser:

The polar boundaries are r=1..sqrt(5) and theta=0..2*Pi Also, x^2=(r*cos(theta))^2 and y=r*sin(theta):

Thus the volume is given by the following integral:

>	int(int(((r*cos(theta))^2+(r*sin(theta)))*r,r=1..sqrt(5)),theta=0..2*Pi);

2. Use change of variables to find the volume of the solid region bounded above by the hemisphere  z=sqrt(16-x^2-y^2) and below by the circular region R given by x^2+y^2 is less than or equal to 4.

Anwser:

The region x^2+y^2 is less than or equal to 4 is simply a circle in the xy plane with a r=2. Thus the bounds would be r=0..2 and y=0..2Pi In addition, the
hemisphere which forms the top of the cylinder can be changed into polor coordinates: z=sqrt(16-x^2-y^2) --> z=sqrt(16-r^2)

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

>	int(int(sqrt(16.0-r^2)*r,r=0..2),theta=0..2*Pi);