Thread: Find centroid of the following double integral

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.

Originally Posted by damule

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).

The mass of the region



with density is



and the centroid is



Why do you want to change the order of integration? Have you had any computation difficulties?

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?

Originally Posted by damule

How did they find that m? On what do you rely? I hope someone could explain to me in further detail

is the area of the region . If (constant) the mass of is ( area of , which is a sector). You can choose for finding the centroid ( appears multiplying in the numerator and denominator and can be canceled).