#### 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=8π unit3

#### Explanation

**1) Concept:**

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

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

V= ∫ab2πy f(y)dy

where, 0≤a≤b

**2) Given:**

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

**3) Calculation:**

As the region is bounded by y=x ⇒x=y2, x=0 y=2

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

Therefore, the circumference is 2πy and the height is x=y2

a=0 and b=2

So, the total volume is

V= ∫ab2πy [y2] dy

V= ∫022πy y2dy

V=2π∫02y3dy

V=2πy4402

=2π 244-0

=2π4

V=8π

**Conclusion:**

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

V=8π unit3