Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
x = 5, y = 0; about the y-axis
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
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
Finally the actual volume we want is obtained by subtracting the above two volumes
= 625π/16 – 625π/32 = 625π/32
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