How to Determine Moment of Inertia by Integration: Step-by-Step Guide

[drunknfox]

Homework Statement

Determine by direct integration the moment of inertia of the shaded area with respect to the y axis

http://imgur.com/O0Fu6

Homework Equations

Ix = &int y^2 dA Iy= &int x^2 dA

The Attempt at a Solution

I don't know how to start this problem. some practice examples use y=b x=a but that does not work because the parenthesis turn to 0. I just need help starting the problem

 

[jambaugh]

I_y = \int x^2 dA
What is dA? dA = area of the slice you are adding up, probabily dA = [f(x) - 0]dx since you are given a region bound by y=f(x) and y=0.

In general, you'll integrate over the region and you must determine your variable of integration.

You then express the variables in the formula in terms of the independent variables.

You'll then do the integration.

As a rule you will typically find it best to integrate with respect to the variable(s) orthogonal to the axis of rotation for a given moment of inertia. The reason being, as you set up that integral via Riemann sums, the slices will consist of points all the same distance from the axis of rotation.

These formulas derive from the moment of inertia in a more general context. Given a region with unit density and with some defined axis or rotation, then the moment of inertia about that axis is the sum of the moments of inertia of all the pieces when you chop it up via Riemann sums.

I_{axis} = \int dI
where dI is the moment of inertia of each infinitesimal piece.

If you're slicing parallel to the axis then dI = r^2 times dm where dm is the differential mass, (typically density times differential area or volume hence = dA in your case)
and r is the distance of that piece from the axis.

If you're slicing perpendicular to the axis then dI will be the formula for the moment of inertial of a stick of infinitesimal thickness swung about an axis and I won't go into it. Pick slicing parallel to the axis and hence integrate w.r.t the variable running perp. to the axis, or wait until Calc III when you can do a multiple integral, i.e. dice the region up into point sized pieces.