Is my procedure correct to compute this moment of inertia J?

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SUMMARY

The discussion focuses on the calculation of the moment of inertia J for an object revolving around the y-axis, specifically using the function z² = (r²/a) y. The correct formula for the moment of inertia, as concluded by the participants, is (2 m r²) / 3, which does not include the variable "a". Participants identified errors in the initial calculations and confirmed the correct approach to derive the moment of inertia based on uniform density and the specified limits of y from 0 to "a".

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Marvin94
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Looking at the image below, we have the function

z^2=\frac{r^2}{a} yrevolving about y-axis. We know that y goes from 0 to "a". We also know the mass of the object (of uniform density). Do you think is my procedure correct? Do you get the same result?

IMG_20150729_183245.jpg
 
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I did wrong calculation. The results should be

(2 m r^2) / 3

without "a".
 

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