#### To determine

**To find:**

The volume generated by rotating the region bounded by the given curves about x=5 using the cylindrical shells.

#### Answer

V=136π15unit3

#### Explanation

**1) Concept:**

i. If x is the radius of the typical shell, then the circumference=2πx and the height is y=f(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= ∫ab2πxf(x)dx

where, 0≤a<b

**2) Given:**

y=x, x=2y; about x=5

**3) Calculation:**

As the region is bounded by y=x, x=2y; about x=5, draw the region using the given curves.

The graph shows the region and the height of typical cylindrical shell formed by the rotation about the line x=5

It has the radius =5-x

Therefore, the circumference is 2π(5-x) and the height is x-12x.

To find the point of intersection, equate the curves that is

x=12x

Thus

x=0, 4

So, a=0 and b=4

So, the volume of the given solid is

V= ∫ab2π(5-x)fxdx

=∫042π5-xx-12xdx

=2π∫045x12-52x-x32+12x2dx

=2π5x3232-52·x22-x5252+12·x3304

=2π10x323-5x24-2x525+x3604

=2π104323-5424-24525+436-0

=2π6815

=136π15

**Conclusion:**

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

V=136π15unit3