A lamina occupies the region inside the circle x^2 + y^2 = 6x,
but outside the circle x^2 + y^2 =9.
Find the center of mass if the density at any point is inversely
proportional to its distance from the origin.
A lamina with constant density
ρ(x, y) =
ρ
occupies the given region. Find the moments of inertia
Ix
and
Iy
and the radii of gyration and .
The part of the disk
x2 +
y2 ≤
a2
in the first quadrant
Find the mass and center of mass of the lamina that occupies the
region D and has the given density function ρ.
D is bounded by y = 1 − x2 and y = 0; ρ(x, y) =
5ky
m=
(x bar ,y bar)=
Let R be the region inside x ^2 /9 + y^2 /25 = 1, with x ≥ 0.
(That is, R is the right-hand side of the ellipse below.) (a) (15
pts) Use the change of variables x = 3u, y = 5v to transform ∫ ∫ R
x dxdy to a polar coordinate integral. 3 −5 5 R (b) (5 pts)
Evaluate the polar coordinate integral
A lamina occupies the part of the rectangle 0≤x≤40≤x≤4,
0≤y≤30≤y≤3 and the density at each point is given by the function
ρ(x,y)=3x+7y+4ρ(x,y)=3x+7y+4.
A. What is the total mass?
B. Where is the center of mass?
(18) The region is bounded by y = 2 − x 2 and y = x.
(a) Sketch the region.
(b) Find the area of the region.
(c) Use the method of cylindrical shells to set up,
but do not evaluate, an integral for the volume of the solid
obtained by rotating the region about the line x = −3.
(d) Use the disk or washer method to set up, but do
not evaluate, an integral for the volume of...
A solid S occupies the region of space located outside the
sphere x2 + y2 + z2 = 8 and inside
the sphere x2 + y2 + (z - 2)2 = 4.
The density of this solid is proportional to the distance from the
origin.
Determine the center of mass of S.
Is the center of mass located inside the solid S ?
Carefully justify your answer.