http://www.dj1love.com/axis-equipment-2/
Can someone PLEASE tell me what this question means and solve for it? CALC 3 double integral problem…?
A 2-dimensional region exists with a boundary given by:
y=sqrt(9-x^2), y= -sqrt((1/4)-x^2),
and the two line segments along the x-axis needed to close the region. Consider this region a 2-dimensional piece of material that needs analyzing so a decision can be made whether to use it or not in a certain manufacturing situation. The piece of equipment being manufactured has mass 20,000 kg without the 2-dimensional piece of material being discussed here. The total mass, including the 2-dimensional piece of material must be less than 60,000 kg. The density of the 2-dimensional piece of material is variable, and is equal to 400 times the distance of a point of the material to the origin. Determine whether the 2-dimensional piece of material can be used or not, and find the point at which it is horizontally stable.
please give as much detail as possible
the upper part is a circle of radius 3
the lower part is a circle with radius 1/2
the density is 400 √(x^2 + y^2)
but to get the total mass, expressing it in polar form is easier
M = ∫ ∫ density dx dy
= ∫ ∫ density r dr dθ
= ∫ (0 , π) ∫ (0 , 3) 400 r * r dr dθ +
∫ (π , 2π) ∫ (0 , 1/2) 400 r * r dr dθ
= 400π r^3/3 (when r = 3) + 400π r^3/3 (when r = 1/2)
= 11,362
thus the material can still be used
to determine the horizontal stability, we need
Mx = ∫ ∫ y * density dx dy
here, you need to incorporate “r sin&theta” in the integrand.
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