The accompanying figure on the next page shows the solid that is bounded above by the surface z = 1 / x 2 + y 2 + 1 , below by the xy -plane, and laterally by the surface x 2 + y 2 = a 2 . (a) By symmetry, the centroid of the solid lies on the z -axis. Make a conjecture about the behavior of the z -coordinate of the centroid as a → 0 + and as a → + ∞ . (b) Find the z -coordinate of the centroid, and check your conjecture by calculating the appropriate limits. (c) Use a graphing utility to plot the z -coordinate of the centroid versus a , and use the graph to estimate the value of a for which the centroid is 0 , 0 , 0.25 .
The accompanying figure on the next page shows the solid that is bounded above by the surface z = 1 / x 2 + y 2 + 1 , below by the xy -plane, and laterally by the surface x 2 + y 2 = a 2 . (a) By symmetry, the centroid of the solid lies on the z -axis. Make a conjecture about the behavior of the z -coordinate of the centroid as a → 0 + and as a → + ∞ . (b) Find the z -coordinate of the centroid, and check your conjecture by calculating the appropriate limits. (c) Use a graphing utility to plot the z -coordinate of the centroid versus a , and use the graph to estimate the value of a for which the centroid is 0 , 0 , 0.25 .
The accompanying figure on the next page shows the solid that is bounded above by the surface
z
=
1
/
x
2
+
y
2
+
1
,
below by the xy-plane, and laterally by the surface
x
2
+
y
2
=
a
2
.
(a) By symmetry, the centroid of the solid lies on the z-axis. Make a conjecture about the behavior of the z-coordinate of the centroid as
a
→
0
+
and
as
a
→
+
∞
.
(b) Find the z-coordinate of the centroid, and check your conjecture by calculating the appropriate limits.
(c) Use a graphing utility to plot the z-coordinate of the centroid versus a, and use the graph to estimate the value of a for which the centroid is
0
,
0
,
0.25
.
Precalculus Enhanced with Graphing Utilities (7th Edition)
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