Chapter 5 - Applications Of The Definite Integral In Geometry, Science, And Engineering - 5.3 Volumes By Cylindrical Shells - Exercises Set 5.3 - Page 370: 5

$$V = \frac{{2\pi }}{5}$$

$$\eqalign{ & {\text{We have }}y = {x^3},\,\,\,x = 1,\,\,\,\,y = 0 \cr & {\text{The volume of the solid can be calculated using cylindrical shells}} \cr & V = \int_a^b {2\pi x\left[ {f\left( x \right) - g\left( x \right)} \right]dx} \cr & {\text{Let }}f\left( x \right) = {x^3}{\text{ and }}g\left( x \right) = 0 \cr & V = \int_0^1 {2\pi x\left( {{x^3} - 0} \right)dx} \cr & V = 2\pi \int_0^1 {{x^4}dx} \cr & {\text{Integrating}} \cr & V = 2\pi \left[ {\frac{1}{5}{x^5}} \right]_0^1 \cr & V = \frac{{2\pi }}{5}\left[ {{x^5}} \right]_0^1 \cr & V = \frac{{2\pi }}{5}\left( 1 \right) \cr & V = \frac{{2\pi }}{5} \cr} $$