the line x = 4, Locate the centroid of the resulting solid of revolution in 500 words
The centroid of a solid of revolution can be determined using the formula C = (π/A) ∫ xydA, where A is the cross-sectional area and dA is an elemental area. We first need to determine the bounds for our integral which will be between 0 and 4 according to the problem description. Now we must calculate the cross-sectional area of this region in order to solve for C. Since we are revolving about the line x = 4, all points on this boundary will have y = 0. Thus, integrating from y = 0 to y = 4 gives us A = π∫0x2dx4 , so that A= 32π . Therefore, our centroid equation becomes:
To solve this integral we use integration by parts with u=xy and dv= dy:
Therefore, ∫udvydy0x24dy=([xy]40+2∫22y2dx)-(30+(2∫21)()*12*3)=60+8π =>C=(1/32π)(60+8π)= 5/6
Therefore,the centroid of the resulting solid of revolution is located at (5/6 ,0).
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