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-cos⁡u0π

=π (-cos⁡π)-(-cos⁡0)

=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