Question

To find: The volume generated by rotating the region bounded by the given curve by the use of the method of cylindrical shell.

Short answer

The volume generated by rotating the region bounded by the given curve by the use of the method of cylindrical shell is \(\frac{{512}}{5}\pi \).

Step-by-step solution

Given information

The function is \(x = 4{y^2} - {y^3},x = 0\).

The concept of volume by cylindrical shell method

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\(a\)to\(b\)is\(V = \int_a^b 2 \pi xf(x)dx\).

Calculate the volume

It is given that the curves that bound a region are \(x = 4{y^2} - {y^3},x = 0\).

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

With\(x = 0\), the value of\(y\)is as shown below.

\(\begin{aligned}{}4{y^2} - {y^3} = 0\\{y^2}(4 - y) = 0\\{y^2} = 0,4 - y = 0\\y = 0,y = 4\end{aligned}\)

Here, the shell radius is\(y\)and the shell height is\(4{y^2} - {y^3}\).

The region lies between\(y = 0\), and\(y = 4\), so by definition of cylindrical shell, the volume becomes

\(\begin{aligned}{}V = \int_0^4 2 \pi y\left( {4{y^2} - {y^3}} \right)dy\\ = 2\pi \int_0^4 4 {y^3} - {y^4}dy\\ = 2\pi \left( {{y^4} - \frac{{{y^5}}}{5}} \right)_0^4\\ = \frac{2}{5}\pi \left( {5{{(4)}^4} - {{(4)}^5}} \right)\end{aligned}\)

That is, \(V = \frac{{512}}{5}\pi \).