How do you find the centroid of this?

[Manwe Sulimo]

Homework Statement

Find the centroid of the shape formed by the equation y²=x³-x⁴, the x-axis, and the y-axis.

Homework Equations

A=∫f(x)dx
Mx=∫(1/2)[f(x)]²dx
My=∫x[f(x)]dx

The Attempt at a Solution

I'm stuck on the integral.
I attempted u-substitution and got du=(1/2)(x³-x⁴)^-1/2dx; "parts," trigonometric substitution/identities, and partial fractions don't seem to apply.

 

[SammyS]

Homework Statement

Find the centroid of the shape formed by the equation y²=x³-x⁴, the x-axis, and the y-axis.

Homework Equations

A=∫f(x)dx
Mx=∫(1/2)[f(x)]²dx
My=∫x[f(x)]dx

The Attempt at a Solution

I'm stuck on the integral.
I attempted u-substitution and got du=(1/2)(x³-x⁴)^-1/2dx; "parts," trigonometric substitution/identities, and partial fractions don't seem to apply.

Try polar coordinates.

 

[Manwe Sulimo]

Try polar coordinates.

I haven't worked in those for a while (I'm not sure I remember how to use them); but I'm fairly certain my teacher wants me to stick with x-y coordinates.

 

[SteamKing]

I haven't worked in those for a while (I'm not sure I remember how to use them); but I'm fairly certain my teacher wants me to stick with x-y coordinates.

You've already tried trig substitutions. Think of polar coordinates as a form of trig substitution.

Besides, what will your teacher prefer? That you kept working with cartesian coordinates and didn't solve the problem, or you converted to polar coordinates and got an answer?

 

[vela]

Homework Statement

Find the centroid of the shape formed by the equation y²=x³-x⁴, the x-axis, and the y-axis.

Homework Equations

A=∫f(x)dx
Mx=∫(1/2)[f(x)]²dx
My=∫x[f(x)]dx

You swapped $M_x$ and $M_y$.

The Attempt at a Solution

I'm stuck on the integral.
I attempted u-substitution and got du=(1/2)(x³-x⁴)^-1/2dx; "parts," trigonometric substitution/identities, and partial fractions don't seem to apply.

Try starting like this:
$$\int x\sqrt{x^3-x^4}\,dx = \int x^2\sqrt{x-x^2}\,dx = \int x^2\sqrt{\frac 14-\left(x-\frac 12\right)^2}\,dx.$$ The last step comes from completing the square. Then try a few more substitutions and see if you get anywhere.