To determine

To find:

The volume generated by rotating the region bounded by the given curves about y-axis using the cylindrical shells.

Answer

6π7 unit3

Explanation

1) Concept:

i. If x is the radius of the typical shell, then the circumference =2πx.

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=x3   y=0, x=1  rotated about the y-axis.

3) Calculation:

As the region is bounded by y=x3 , y=0, x=1  rotated about the y-axis,

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

Therefore, the circumference is 2πx and the height is y=x3

So, the total volume is

V= ∫ab2πxx3  dx

V= ∫012πxx3dx

V= ∫012π x43dx

V=2π∫01x43dx

V=2π37x7301

=2π 37173-(0)

=2π37

V=6π7

Conclusion:

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

V=6π7 unit3