How can I integrate and find the moment of inertia for a circle?

[ShawnD]

In school, I'm doing this thing called moment of inertia given by the formula

$$I = \int y^2 dA$$

If the object being solved for is a rectangle where the base of it is parallel to the x axis, dA is equal to xdy. From there, the integration is easy. If the object to solve the integration for is a circle (or anything with slants like a triangle), I don't know how to do it. How do I solve for dA of a circle?

 

[himanshu121]

Consider circular rings with inner and outer radius r and r+dr Then dA=rdr

 

[HallsofIvy]

"dA" is the "differential of area". If you have a circle centered at the origin, then polar coordinates are natural and dA= r dr dθ.

In Cartesian coordinates dA= dxdy. That can be used when you are talking about a circle but the calculations for the limits of integration will be more complicated.

By the way, do you understand that the formula you give is specifically for the moment of inertia when rotating around the x-axis?

In polar coordinates, y= r cosθ so the moment of inertia of a disk, of radius R, centered at (0,0) and rotated around the x-axis is:

∫_θ=0^2π∫_r=0^Rr²cos²(r dr dθ)