What is the Correct Moment of Inertia for a Unit Density Lamina?

[longrob]

Homework Statement

A lamina of unit density consists of the region between the two curves $$y=\sqrt{4-x^2}$$ and $$y=1-4x^2$$ and the x axis.
Find it's moment of inertia about the x-axis.

Homework Equations

This is the correct answer:
$$2\left \{ \int_{0}^{2}\int_{0}^{\sqrt{4-x^2}}}y^2 dy dx - \int_{0}^{1/2}\int_{0}^{1-4x^2}y^2 dy dx \right \}$$
which I do understand.

The Attempt at a Solution

$$2\int_{0}^{2}\int_{1-4x^2}^{\sqrt{4-x^2}}y^2 dy dx$$
I don't understand why this is wrong.

Thanks in advance.

 

[Dick]

Because you only want the moment of inertia BETWEEN the two curves. 1-4x^2 is negative for x>1/2. Sketch a graph of the region you want. You have to split the integral into two parts.

 

[longrob]

Thank you. That makes perfect sense !