Mass of a lamina?

Question

A lamina occupies the part of the disk x2+y2≤25 in the first quadrant and the density at each point is given by the function ρ(x,y)=3(x2+y2) .

A. What is the total mass?

B. What is the moment about the x-axis, ∫xρ(x,y)dxdy ?

C. What is the moment about the y-axis, ∫yρ(x,y)dxdy ?

D. Where is the center of mass? ( , )

E. What is the moment of inertia about the origin, ∫(x2+y2)ρ

Sorry, I didn't realize the second powers didn't come through right. It should say:
A lamina occupies the part of the disk x^2+y^2≤25 in the first quadrant and the density at each point is given by the function ρ(x,y)=3(x^2+y^2) .

A. What is the total mass?

B. What is the moment about the x-axis, ∫xρ(x,y)dxdy ?

C. What is the moment about the y-axis, ∫yρ(x,y)dxdy ?

D. Where is the center of mass? (, )

E. What is the moment of inertia about the origin, ∫(x^2+y^2)ρ(x,y)dxdy ?

Answer 1

Too many questions, I'll help you with the 1st one. The elementary mass is dm = p(x,y) dx dy and m is the integral of this expression over the 1st quadrant of radius 5. To solve this, I think it's easier to use polar coordinates, so that

x = r cos(t), y = r sin(t) and dx dy = r dr dt. Since we're integrating over the 1st quadrant, t varies from 0 to 5 and t from 0 to pi/2. And since x^2 + y^2 = r^2,

m = Int (0 to 5) Int(0 to pi/2) 3 r^2 r dr dt = 3pi/2 Int (0 to 5) r^3 dr = 3pi/2 [r^4/4] {from o to 5] = 3pi/2 (625/4) = 234.375 pi units of mass.

The other questions can be done in a similar way.

Thenn, m = In