MULTIPLE INTEGRALS AND THEIR APPLICATIONS
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where symmetry was again invoked in the last step. Change to polar coordinates and use Problem 29.9:
Find the moment of inertia of a cube of edge a with respect to an edge if the density is numerically equal to the
square of the distance from one extremity of that edge.
Consider the cube
Let the density be
the square of the
distance from the origin. Let us calculate the moment of inertia around the *-axis.
[The distance from (x, y, z) to the ;t-axis is
Then
44.89
44.88
Find the centroid of the region outside the circle r = 1 and inside the cardioid r—\ + cos 0.
Refer to Fig. 44-20. Clearly, y = 0, and x is the same for the given region as for the half lying above the
polar axis. For the latter, the area
The moment about the y-axis is
44.90
Find the centroid (x, y) of the region in the first quadrant bounded by y
2 = 6x, y = 0, and x = 6 (Fig.
44-43).
The area
The moment about the
*-axis is
Hence,
The moment about the y-axis is
So the centroid is
Hence,
Fig. 44-43
44.91
Find the centroid of the solid under z
2 = xy and above the triangle bounded by y = x, y = 0, and x = 4.
The volume
The moment about the _yz-plane is
Hence,
The moment about the