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Unformatted text preview: dz dr dθ = m=
3 2π =
0 0
2π =
0 0
4 12
z r + r2 z dz dr dθ =
2 3 8r + 4r2 dr dθ
0 0 0 3 0 0 2π 0 0 0 2π 2π 4
4r2 + r3 dθ =
3 = 144π 36 + 36 dθ = 72θ
0 0 0 10. Let Q be the tetrahedron bounded by the coordinate planes and the plane 2x + 5y + z = 10. Find the mass and center
of mass of Q is the density at a point P (x, y, z ) is directly proportional to the distance from P to the xz plane.
First, notice that the distance of a point from the xz plane is given by y . Since the volume Q is a subset of the ﬁrst
octant y is always positive, thus δ (x, y, z ) = ky for some constant of proportionality k .
Next, notice that if we decompose Q with respect to z , then the top surface is given by z = 10 − 2x − 5y and the bottom
surface is given by z = 0.
The region R in the plane that deterines the outer pair of limits of integration in our triple integrals is a triangle in
the ﬁrst quadrant of the xy plane. This triangle is bounded above by y = 1 (10 − 2x) [set z = 0 in the equation for the
5
plane and then solve for y ] and below by y = 0.
Finally, we see that 0 ≤ x ≤ 5 [notice tha xintercept of the plane is at x = 5.] From this, we can set up triple integrals for the mass and for each of the three moments:
1
5 (10−2x) 5 m=
0 0 10− 2 x − 5 y ky dz dy dx =
0 0
1
5 (10−2x) 5 =k
0 0
5 0
5 10 − 2x
5 =k
0 −x
2 10 − 2x
5 5−x
dx = k
3 2 −
5
0 5
3 0 dy dx = kyz 0 5
5y 2 − xy 2 − y 3
3 10 − 2x
5 1
5 (10−2x) 5 0 10y − 2xy − 5y 2 dy dx = k
2 10− 2 x − 5 y 0
5 10 − 2x
5 5 =k 1
5 (10−2x) 5 3 5 dx = k
0 0 10− 2 x − 5 y ky (10−2x−5y ) 1
5 (10−2x) dx
0 10 − 2x
5 4
1
(5 − x)3 dx = k (5 − x)4
75
75 5 =k
0 2 5−x− 5
3 25
625
=
k
75
3 Similarly, (omitting the details of the evaluations, which are very much like the one above):
1
5 (10−2x) 5 Myz =
0 1
5 (10−2x) 5 Mxz =
Mxy =
0 Thus x = 0
Myz
m 25
k
3 10− 2 x − 5 y yky dz dy dx = 20
k
3 zky dz dy dx = 50
k
3 0 0
5 xky dz dy dx = 0 0 0 10− 2 x − 5 y 1
5 (10−2x) 10− 2 x − 5 y
0 = 1, y = Mxz
m = 4 , and z =
5 Mxy
m = 2. dy dx...
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This document was uploaded on 03/11/2014 for the course MATH 323 at Minnesota State University Moorhead .
 Fall '11
 James
 Math, Vector Calculus

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