#### To determine

**To find:**

The volume generated by rotating the region bounded by the given curves about the x-axis using the cylindrical shells.

#### Answer

V=192π unit3

#### Explanation

**1) Concept:**

i. If y is the radius of the typical shell, then the circumference =2πy .

ii. By the shell method the volume of the solid by rotating the region under the curve y=f(x) about the x- axis from a to b is

V= ∫ab2πyf(y)dy

where 0≤a≤b

**2) Given:**

The region bounded by y=x32, x=0, y=8 rotated about the x- axis.

**3) Calculation:**

As the region is bounded by

y=x32 ⇒x=y23, x=0, y=8

Using the shell method, find the typical approximating shell with the radius y

Therefore, the circumference is 2πy and the height is

x=y23

a=0 and b=8

So, the total volume is

V= ∫ab2πy [y23] dy

V= ∫082πy y23dy

V=2π∫08y53dy

V=2π3y83808

=2π 38883-0

=2π38·256

V=192π

**Conclusion:**

The volume of the solid obtained by rotating the region bounded by the given curves is

V=192π unit3