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Home Multiple Integrals Cylindrical and Spherical Coordinates Examples Example 1
 
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Example 1

Find the moment of inertia of a cylinder of height h, base a circle of radius b, and constant density 1, about its axis.

Step 1

Draw the region as in Figure 12.7.9.

Step 2

The problem is greatly simplified by a wise choice of coordinate axes. Let the z-axis be the axis of the cylinder and put the origin at the center of the base. Then the region E in rectangular coordinates is

-b ≤ x ≤ b, -12_multiple_integrals-400.gif≤y≤12_multiple_integrals-401.gif, 0≤z≤h, and in cylindrical coordinates is 0 ≤ 0 ≤ 2π, 0≤r≤b, 0 ≤ z ≤ h.

Step 3

The problem looks easier in cylindrical coordinates.

x2 + y2 = r2.

12_multiple_integrals-402.gif

12_multiple_integrals-403.gif

Figure 12.7.9
Home Multiple Integrals Cylindrical and Spherical Coordinates Examples Example 1

Last Update: 2006-11-15