Moment of inertia using double integration.

Question

Need to find moment of inertia about $x$-axis using double integration if density at $(x,y) =y+1$ of area bounded by $x=2y-y^2$ and $x=y^2$. My difficulty arises in converting $x=2y-y^2$ to $y=?$ as one of the limits in the integral.

Answer 1

You can of course solve $x=2y-y^2$ for $y$, this is a simple quadratic equation, but do you have to? Just switch your order of integration: $$\int_0^1\int_{2 y - y^2}^{y^2} f(x,y) dx dy$$ Note $y^2=2 y-y^2$ at $y=0,y=1$, so $0,1$ are the bounds for $y$.