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Solve for the Volume of the Solid

label Mathematics
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Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

y =1/16x^2 

x = 5,  y = 0;    about the y-axis

Oct 17th, 2017

First find the intersection of the curve and the line x= 5:

When x=5,  becomes I/16 * 5^2 =25/16

The lower limit for the revolution about the Y-axis is y= 0 and the upper limit is y=25/16

Now the volume obtained by revolving about the Y-axis the lines bounded by  x= 5, y=0 and y=25/16 is simply a cylinder with radius 5 and height 25/16 is given by π (5^2) * 25/16

=625π/16

The  volume obtained by  revolving about the Y-axis the area bounded by the curve y=1/16 x^2,  y=25/16 (it is a solid cup-shape)  is given by π ∫ x^2 dy 

=  π ∫ 16y dy   (multiplying  both sides of y= 1/16 x^2 by 16)

8 π [y^2] from limits y=0 to y= 25/16

=8π * (25/16)^2

=625π/32

Finally the actual  volume we want is obtained by subtracting the above two volumes

 = 625π/16 – 625π/32 = 625π/32


Sep 9th, 2014

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