#### To determine

**To find:**

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

#### Answer

V=8π unit3

#### Explanation

**1) Concept:**

i. If x is the radius of a typical shell, then the circumference =2πx and the height is y

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

V= ∫ab2πxf(x)dx

where 0≤a≤b

**2) Given:**

The region bounded by y=x2, y=4, 0≤ x≤2, x=0 rotated about the y-axis.

**3) Calculation:**

As the region is the bounded by y=x2, y=4, 0≤ x≤2, x=0 rotated about the y-axis

for shell method, the typical approximating shell with the radius x is

Therefore, the circumference is 2πx and the height is y=4-x2.

So, the total volume is

V= ∫ab2πx[4-x2] dx

V= ∫022π[4x-x3]dx

V=2 π∫02[4x-x3]dx

V=2π4x22-x4402

=2π 2·22-244-0

=2π8-4

V=8π

**Conclusion:**

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

V=8π unit3