#### To determine

**To find:**

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

#### Answer

2π unit3

#### 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=sin(x2) rotated about the y-axis.

**3) Calculation:**

As the region is bounded by y=sin(x2) rotated about the y-axis,

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

Therefore, the circumference is 2πx and the height is y=sin(x2).

So, the total volume is

V= ∫ab2πx sin(x2)dx

V= ∫0π2πx sin(x2)dx

u=x2⇒2xdx=du

x=0⇒u=0 and x=π⇒u=π

Therefore,

V= ∫0ππ sin(u)du

V= π∫0πsin(u)du

V=π-cosu0π

=π (-cosπ)-(-cos0)

=π1-(-1)

=2π

V=2π

Let us consider the problem of finding the volume by the washer method

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

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

y=sin(x2) for x in terms y.

By finding the functions x=g1y and x=g2y, we would finally find the volume using

V=g2y2 -g1y2.But to solve for x in terms y is difficult.

Therefore, using the shells is definitely preferable to slicing.

**Conclusion:**

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

V=2π unit3