Multiple Choice Question 9.4. Locate the center of gravity of the volume generat

Question

Multiple Choice Question 9.4.

Locate the center of gravity of the volume generated by revolving the shaded area about the z axis. The material is homogeneous.

I got the right answer from another chegg question post; but I want to understand why the answer involves dV= pi * y^3 * dz; and not dV= pi * y^2 * dz since I'm assuming that r=y and volume of a cylinder (the thin disk) is volume= pi * r^2 * height. The other post had this,

which gives the right answer of Z bar equals 2.67. Why is r^2=y^3 in this question???

Multiple Choice Question 9.4 Part A Locate the center of gravity of the volume generated by revolving the shaded area about the z axis. The material is homogeneous. 4 ft · Z-2.67 ft Z=2.50 ft Z 3.00 ft Z = 2.80 ft 4 ft 42 Submit My Answers Give Up Correct

Explanation / Answer

Y is not the radius. It is the equation of the Shape. Equation of the shape should be integrated to get the volume of the desired shape. This is not a cylinder. For cylinder you can directly apply the formula as you said above. But this is the volume which generated by rotating a parabola about y axis and equation of which is given. So, we need to integrate to get the volume for irrregular shapes like this.

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