Question

To determinethe volume generated by rotating the region bounded by the given curve about y- axis.

Short answer

The volume of the solid is \(\frac{{768}}{7}\pi \).

Step-by-step solution

Given

The region bounded by the given curve about y- axis.

The Concept ofvolume by the use of the method of shell

The volume by the use of the method of shell is\(V = \int_a^b 2 \pi x(f(x))dx\)

Evaluate the volume

Definition of Volume by cylindrical shell method:

Consider that the solid \(S\) is obtained by rotating the region under the curve \(y = f(x)\) about the \(y\)-axis from \(o\) to \(b\) is \(V = \int_a^b 2 \pi xf(x)dx\).

It is given that the curves that bound a region are \(y = {x^3},y = 8,x = 0\).

The specific line around which the region is rotated is \(x\)-axis.

With \(y = 8\), the value of \(x\) becomes \(x = 2\).

Here, the shell radius is \(y\) and the shell height is \({y^{\frac{1}{3}}}\).

The region lies between \(y = 0\), and \(y = 8\), so by definition of cylindrical shell, the volume becomes

\(\begin{aligned}{}V = \int_0^8 2 \pi y\left( {{y^{\frac{1}{3}}}} \right)dy\\ = \int_0^8 2 \pi {y^{\frac{4}{3}}}dy\\ = 2\pi \left( {\frac{3}{7}{y^{\frac{7}{3}}}} \right)_0^8\\ = \frac{6}{7}\pi {(8)^{\frac{7}{3}}}\end{aligned}\)

That is, \(V = \frac{{768}}{7}\pi \).

Therefore, the volume of the solid is \(\frac{{768}}{7}\pi \).