#### To determine

**To find:**

The volume of the solid obtained by rotating the region shown in the figure about the y-axis and sketch the typical approximating shell.

#### Answer

π15unit3

#### 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=xx-12 rotated about the y- axis.

**3) Calculation:**

As the region is bounded by y=xx-12 rotated about the y-axis,

let us consider the problem of finding the volume by the washer method.

If we slice perpendicular to y-axis then we get a washer.

To compute the inner radius and the outer radius of the washer, we would have to solve the cubic equation.

y=xx-12 for x in terms y that’s not easy.

that is, to find the functions g1and g2 as shown in fig is different.

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

Therefore, the circumference is 2πx and the height is y=xx-12

So, the total volume is

V= ∫ab2πx xx-12dx

V= 2π∫01x4-2x3+x2dx

By using the fundamental theorem of calculus and the power rule of integration,

V=2π15x5-2x44+x3301

=2π 15·15-142+133-0

=2π15-12+13

=2π 6-15+1030 =2π30

V=π15

**Conclusion:**

The volume of the solid obtained by rotating the region bounded by the given curves is V=π15unit3