Question

onsider the solid obtained by rotating the region bounded by the given curves about the y-axis. y = x^2/16 text(, ) x = 2 text(, ) y = 0 Find the volume V of this solid.

Answer 1

The volume of solid will be
`(\pi)/(2)` cubic unit

Explanation:

Given: The given curves
`y=(x^2)/(16)`

Rotation about y-axis to form a solid bounded by given curve, x=2 and y=0.

Please see the attachment for figure.

Volume of solid rotation about y-axis using cylindrical shell method.

`V=\int_a^b2\pi rhdx`

where,

a is lower limit (a=0)

b is upper limit (b=2)

r is radius (r=x)

h is height (
`h=y=(x^2)/(16)`)

using the above formula the volume of solid we get

`V=\int_0^22\pi\cdot(x^3)/(16)dx`

`V=2\pi\cdot(x^4)/(64)|_0^2`

`V=(\pi)/(2)`

Hence, The volume of solid will be
`(\pi)/(2)`