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Home Multiple Integrals Double Integrals in Polar Coordinates Examples Example 2
 
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Example 2

Find the mass and center of mass of a flat plate in the shape of a semicircle of radius one whose density is equal to the distance from the center of the circle.

Step 1

The region D is sketched in Figure 12.5.9.

12_multiple_integrals-280.gif

Figure 12.5.9

Step 2

Take the origin at the center of the circle and the x-axis as the base of the semicircle. D is the polar region 0≤6≤π, 0≤r≤1.

Step 3

The density is

12_multiple_integrals-275.gif

12_multiple_integrals-276.gif

Answer 12_multiple_integrals-277.gif, 12_multiple_integrals-278.gif, 12_multiple_integrals-279.gif~ 0.477.

The point (x,y) is shown in Figure 12.5.10.

 12_multiple_integrals-281.gif

                                                           Figure 12.5.10
Home Multiple Integrals Double Integrals in Polar Coordinates Examples Example 2

Last Update: 2006-11-15