Verify the given moment(s) of inertia and find x and y. Assume that each lamina has a density of p = 1 gram per square centimeter. (These regions are common shapes used in engineering.)
Circle
=
=
a
Suppose the solid W in the figure is the spherical half-shell consisting of
the points above the xy-plane that are between concentric spheres
centered at the origin of radii 4 cm and 10 cm. Suppose the density 8 of the
material increases linearly with the distance from the origin, and that at the
inner surface the density is 8 g/cm³ while at the outer surface it is
10 g/cm.
(a) Using spherical coordinates, write d as a function of p. Enter p as rho.
8(e) = 25/9(rho-4)
(5.0
(b) Set up the integral to calculate the mass of the shell in the form below. If
necessary, enter o as phi, and 0 as theta.
B D
CLI 25/9(rho-4)rho^2sinphi
"OP Ópdp
A = 0
B = 2pi
C = 0
D= pi/2
(Drag to rotate)
E- 4
F= 10
(c) Find the mass of the shell.
4536pi
The boundary of a lamina consists of the semicircles
y=
1-x²
and y =
25-x²
together with the portions of the x-axis that join them. Find the center of mass of the lamina if the density at
any point is inversely proportional to its distance from the origin. Hint: use polar coordinates
Precalculus: Mathematics for Calculus - 6th Edition
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