Question

Let R be the region bounded by the following curves. Use the disk or washer method to find the volume of the solid generated when R is revolved about the​ y-axis.

y= square root of (x/2) , y=0 , x=2

Answer 1

3.2 pi

Explanation:

Given 3 curves are:

y = square root ( x / 2)

y = 0

x = 2

Use washer method for hollow volumes.

Step 1: Compute A (y)

A ( y ) = pi * ( f_1 (y) ^2 - f_2 (y) ^2)

where,

f_1 (y) is the function further away from y axis

f_2 (y) is the function closer to y axis

f_1 (y) = 2

f_2 (y) = 2*y^2

A ( y ) = pi * ( 2 ^2 - (2*y) ^2)

A (y) = pi * (4 - 4*y^2)

A (y) = 4*pi * (1 - y^2)

Step 2: Compute V (y)

`V = \int\limits^1_0 {A (y)} \, dy \\V = 4*pi\int\limits^1_0 {1 - y^2} \, dy\\\\V = 4 * pi* (y - 0.2 y^5) \limits^1_0\\\\V = 4*pi*(1 - 0.2)\\\\V = 3.2 pi`

Answer: V = 3.2 pi