Question

To find: The Volume of the solid which is obtained on rotating the region bounded by the given curves about the specified line.

Short answer

The volume of the solid is \(\frac{7}{{15}}\pi \).

Step-by-step solution

Given information

The function is\(y = {x^4},y = 0,x = 1\).

The concept of volume by cylindrical shell method

Definition of Volume by cylindrical shell method:

Consider that the solid\(S\)is obtained by rotation of 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 \(y = {x^4},y = 0,x = 1\).

The specific line around which the region is rotated is\(x = 2\)which is parallel to\(y\)-axis.

With\(x = 1\), the value of\(y\)becomes\(y = 1\).

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

The region lies between\(x = 0\), and\(x = 1\), so by definition of cylindrical shell, the volume becomes

\(\begin{aligned}{}V = \int_0^1 2 \pi (2 - x)\left( {{x^4}} \right)dx\\ = 2\pi \int_0^1 {\left( {2{x^4} - {x^5}} \right)} dx\\ = 2\pi \left( {\frac{{2{x^5}}}{5} - \frac{{{x^6}}}{6}} \right)_0^1\\ = 2\pi \left( {\frac{{2{{(1)}^5}}}{5} - \frac{{{{(1)}^6}}}{6}} \right)\end{aligned}\)

That is,\(V = 2\pi \left( {\frac{{12 - 5}}{{30}}} \right) = \frac{7}{{15}}\pi \).

Therefore, the volume of the solid is \(\frac{7}{{15}}\pi \).