Update: must modify λ

My normal trick is to integrate half of the object, then multiply the result by 2 for the final answer. OOOOPS. That won't work very well for the original λ(x) on the HW 8 PDF. It is negative value mass density on the interval (-L/2,0). NO GO!

So we will have to use a modified λ, specifically λ(x) = α|x|, i.e.,

λ(x) = αx, 0 ≤ x ≤ L/2,
      -αx, -L/2 ≤ x ≤ 0.

Using this modified λ(x), you can use the "integrate half, then ×2" trick. Or, you could split the moment of inertia integral up into two integrals, each with different integrands, using λ(x) above.