I am wondering what exactly is the centroid of $x^2+{(-y^3+1)}^{2/3}=1$. It is a closed implicit shape.
I want to know if solving for the centroid is the same thing as solving for a point with the highest "average radius".
Basically you take a segment from the center point inside $x^2+{(-y^3+1)}^{2/3}=1$, to a point on $x^2+{(-y^3+1)}^{2/3}=1$. That is basically the radius, which you add up all the radius's inside the shape, and divide the number of radius's to get the average.
You could also transform this into polar coordinates to compute the y-coordinate, which in polar coordinates is the radius. The point with the average radius is a way of solving for the center.
Is the centroid the same point as the point with the highest average radius?
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I found the centroid, however, my graphing software sage isn't powerful enough to find the point in $x^2+{(-y^3+1)}^{2/3}=1$. You could find the best answer by taking the values of u and v inside the equation that will produce the greatest area for the below implicit equation above the x-axis.
$$({y\cos(x)+u})^{2}+(-{(y\sin(x)+v)}^{3}+1)^{2/3}=1$$
I used the sage graphing software, and the desmos graphing sofware.
Using integration and approximation I ended up getting the centroid to be (0,.8435938); however, I got the point with the highest average radius to be (0,.833).
I couldn't go farther because sage was powerful enough to give me an accurate area and the time of computation takes too long. I would assume they were different, but proving this would be difficult.