12.3 Double Integrals in Polar Coordinates
Double integrals in Polar Coordinates
Z Z
(3x + 4y 2 )dA;
Example 1) Evaluate
R
where R is the region in the upper half-plane bounded by the circles
x2 + y 2 = 1
and x2 + y2 = 4
Example ) Evaluate the double integral
Z 1 Z p1¡x2
¡1
0
¡ 2
¢
x + y 2 dydx
Example 2) Find the volume of the solid bounded by the plane z=0 and the
2
2
paraboloid z = 1 ¡ x ¡ y :
Example ) Use a double integral to find the area enclosed by one loop of the
r = cos 2µ:
four-leaved rose
Example 3) Find the volume of the solid that lies under the paraboloid z = x2 + y2 ;
above the xy-plane, and inside the cylinder x2 + y2 = 2x:
0
