#### To determine

**To find:**

The volume of the solid generated by rotating the region bounded by the given curves about y-axis by any method

#### Answer

V=8πunit3

#### 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= ∫ab2πx f(x)dx

where, 0≤a≤b

**2) Given:**

The region bounded by y=-x2+6x-8, y=0, rotated about the y- axis.

**3) Calculation:**

As the region is bounded by y=-x2+6x-8, y=0 rotated about the y- axis.

Using the shell method,

The radius is x. So, the circumference is 2πx and the height is -x2+6x-8

From the graph, x=2 and x=4 are x coordinate of the intersection of these two curves.

Therefore, a=2 and b=4

So, the total volume is

V= ∫ab2πx[-x2+6x-8]dx

V= ∫242π[-x3+6x2-8x] dx

V=2 π∫24[-x3+6x2-8x] dx

V=2π-x44+6x33-8x2224

V=2π-444+6433-8422--(2)44+6(2)33-8(2)22

=2π-64+128-64--4+16-16

V=2π(4)

V=8π

**Conclusion:**

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

V=8π unit3