Please explain! Thank you! A solid is generated by revolving the region bounded
ID: 2894932 • Letter: P
Question
Please explain! Thank you!
A solid is generated by revolving the region bounded by the following about the y-axis. y = 1/6 x^2 y = 6 A hole, centered along the axis of revolution, is drilled through this solid so that one-fourth of the volume is removed. Find the diameter of the hole. Consider the graph of y^2 = x(4 - x)^2 (see figure). Find the volumes of the solids that are generated when the loop of this graph is revolved about each of the following (a) the x-axis (b) the y-axis (c) the line x = 4Explanation / Answer
6)
given y=(1/6)x2,y=6
when curves intersect ,
(1/6)x2=6
=>x=6
y axis is x=0
for volume of solid about y axis,using shell method:
height of shell =6-(1/6)x2
radius of shell =x
volume,V=[0 to 6]2x(6-(1/6)x2) dx
volume,V=[0 to 6](/3)(36x-x3) dx
volume,V=[0 to 6](/3)(18x2-(1/4)x4)
volume,V=(/3)(18*62-(1/4)64) -(/3)(18*02-(1/4)04)
volume,V=(/3)(324) -0
volume,V=108
volume of the cylindrical hole =V/4
volume of the cylindrical hole =27
let radius of hole = r ,0<r<6
volume of the cylindrical hole =[0 to r]2x(6-(1/6)x2) dx
=>[0 to r]2x(6-(1/6)x2) dx =27
=>(/3)(18*r2-(1/4)r4) -(/3)(18*02-(1/4)04) =27
=>(/3)(18*r2-(1/4)r4) -0 =27
=>(18*r2-(1/4)r4) =81
=>(72r2-r4) =324
=>(r4-72r2) =-324
=>(r4-72r2)+362 =-324+362
=>(r2-36)2 =972
=>(r2-36) =-183 , (r2-36) =183
=>r2 =36-183 , r2 =36+183=> r>6
=>r2 =36-183
=>r =2.196152422706632
diameter of hole =2r
diameter of hole =2*2.196152422706632
diameter of hole =4.392