we have SS_D f(x,y) dxdy=
I have to first change the order of Integration and then find the centroid (x0,y0) of the region D (the center is the center of mass considering constant mass density).
Maybe its an easy question but i'm just starting to study the subject so I didn't get very far in my attempt to solve this.
I have the following solution which I didn't understand completely (the bold part):
The region is the sector of the circle of radius √(12) centered at the origin between the lines y = 0 and x = y/√(3). You can change the order of integration, but it's a really bad idea from a practical stand point. The order dydx requires two integrals because the upper curve changes. You'd have the limits
0 ≤ x ≤ √(3) and 0 ≤ y ≤ √(3)x for the first integral and
√(3) ≤ x ≤ √(12) and 0 ≤ y ≤ √(12 - x²) for the second integral.
The mass of the region is the area of a sector of a circle of radius √(12) and inscribed angle π/3. This is
m = ½(√(12))² (π/3) = 2π.
The circle and the line y = 3 meet at the point (√(3), 3) and so the inscribed angle Θ satisfies
tanΘ = 3/√(3) = √(3) ==> Θ = π/3.
The moments about the x and y axes are
3 √(12-y²)
∫ ∫ x dx dy = 12 = M_y
0 y/√3
3 √(12-y²)
∫ ∫ y dx dy = 4√(3) = M_x.
0 y/√3
The centroid is (M_y/m, M_x/m) = (6/π, 2√(3)/π).
How did they find that m? On what do you rely? I hope someone could explain to me in further detail
Is there like 2 ways to calculate the mass? Why didn't he do the double integral to find M?