To find the moment of inertia of an object, we take each individual mass element and multiply it by the square of its radius, and then find the sum of all these products.
If the object is continuous, the integral ∫ r^2 dm is used.
If we plot a graph with the y axis being r^2 and the x axis being m, then we get the plot of the graph that we are integrating. In this case, what does the x axis (m) represent?
Best Answer
It's better to think of your integral as a sum at first rather than the area under a curve. The integrand is not a function of $m$ exactly. This can be seen better in the discrete case: $$I=\sum_im_ir_i^2$$ If you wanted to plot each $r_i^2$, the plot would depend on how you label your masses (which mass is $m_1$, which is $m_2$, etc.) So there wouldn't be a unique "plot" to make. $r^2$ isn't a function of $m$, it is a "function" of the index.
When we move to the continuous case the idea is the same. We are just adding up the squared distance from some axis for each mass weighted by the mass of each part. We have a freedom in the "order" in which we add these up. However, in one dimension the typical method involves replacing $\text d m$ with $\lambda \text d x$, where $\lambda$ is the linear mass density. Then you can use an ordering of your masses from smallest to largest $x$ values, where you would then be plotting $\lambda(x)\left(r(x)\right)^2$. Using this method, you could see how the your moment of inertia is the area under this curve.