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

Answer:

The volume of solid will be $\dfrac{\pi}{2}$ cubic unit

Step-by-step explanation:

Given: The given curves $y=\dfrac{x^2}{16}$

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 ($h=y=\dfrac{x^2}{16}$)

using the above formula the volume of solid we get

$V=\int_0^22\pi\cdot\dfrac{x^3}{16}dx$

$V=2\pi\cdot\dfrac{x^4}{64}|_0^2$

$V=\dfrac{\pi}{2}$

Hence, The volume of solid will be $\dfrac{\pi}{2}$