#### 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

62π5unit3

#### Explanation

**1) Concept:**

i. If x is the radius of the 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 y-axis from a to b is

V= ∫ab2πxf(x)dx

where, 0≤a≤b

**2) Given:**

The region bounded by y=x3, y=0, x=1, x=2 rotated about the y-axis.

**3) Calculation:**

As the region is bounded by y=x3, y=0, x=1, x=2 rotated about the y-axis,

Using the shell method, the typical approximating shell with the radius x is shown below,

Therefore circumference is 2πx and height is y=x3

So, the total volume is

V= ∫ab2πxx3 dx

V= ∫122πxx3dx

V= ∫122π x4dx

V=2π∫12x4dx

V=2πx5512

=2π 255-155

=2π325-15

V=62π5

**Conclusion:**

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

V=62π5unit3