Center of mass of area described by implicit function

[Mare102]

Homework Statement

I'm trying to find the center of mass of the region (x²+y²)² =2xy in the first quadrant, but I got stuck.

The Attempt at a Solution

What I did is make the substitution x = r cos(t), y = r sin(t), which gives the equation $$r^{4}=2r²cos(t) sin(t)$$, so r² = sin(2t), so $$r=\sqrt{sin(2t)}$$.
Then the integral for the area of the region becomes (as the Jacobian is r)
$$\int_0^{\pi/2} \int_0^{\sqrt{sin(2t)}} \! r \, dr dt$$

Solving this gives me 0.5.
I’m trying to find the x value of the center of mass, so I want to solve:
$$\int_0^{\pi/2} \int_0^{\sqrt{sin(2t)}} \! r² cos(t) \, dr dt$$
Which gives me
$$\int_0^{\pi/2} \! sin(2t)^{1.5} cos(t) \, dt$$

Which I’m unable to solve. Mathematica gives me a complicated integral, so does anyone know how to proceed, or maybe suggest a different approach?

 

[Mark44]

Homework Statement

I'm trying to find the center of mass of the region (x²+y²)² =2xy in the first quadrant, but I got stuck.

The Attempt at a Solution

What I did is make the substitution x = r cos(t), y = r sin(t), which gives the equation $$r^{4}=2r²cos(t) sin(t)$$,

In the above, what you show is correct, but the exponent on r on the right side renders incorrectly. Use ^ for exponents inside LaTeX expressions. The same problem occurs below, in you integral for M_y (in your calculation for x-bar).

so r² = sin(2t), so $$r=\sqrt{sin(2t)}$$.
Then the integral for the area of the region becomes (as the Jacobian is r)
$$\int_0^{\pi/2} \int_0^{\sqrt{sin(2t)}} \! r \, dr dt$$

Solving this gives me 0.5.
I’m trying to find the x value of the center of mass, so I want to solve:
$$\int_0^{\pi/2} \int_0^{\sqrt{sin(2t)}} \! r² cos(t) \, dr dt$$

The LaTeX expression you want (and mean) is
$$\int_0^{\pi/2} \int_0^{\sqrt{sin(2t)}} \! r^2 cos(t) \, dr dt$$

Which gives me
$$\int_0^{\pi/2} \! sin(2t)^{1.5} cos(t) \, dt$$

You're missing a factor of 1/3 in the integral above. I don't have any good ideas for proceeding here, though.

Which I’m unable to solve. Mathematica gives me a complicated integral, so does anyone know how to proceed, or maybe suggest a different approach?