Question

Problem [2] Find the volume of the solid created by rotating the region enclosed by:y = x and y = x2 about the line x = -1

Answer 1

A solid of revolution is a body that can be obtained by a geometric operation of rotation of a plane surface around a line that is contained in its same plane. In principle, any body with axial or cylindrical symmetry is a solid of revolution.

The first thing we have to do is draw the 2 functions to determine the region to revolutionize:

We will use the definition of volume integral:

$\begin{gathered} V=\int dV \\ V=\int 2\pi\cdot r\cdot h\cdot^{}dx \end{gathered}$

h is the revolution function and r is the displacement in x. In this case

$\begin{gathered} h=x^2 \\ r=-1-x \end{gathered}$

Developing the integral:

$\begin{gathered} V=\int ^0_{-1}2\pi\cdot(-1-x)(x^2)dx \\ V=\int ^0_{-1}2\pi\cdot(-x^2-x^3) \\ V=-\int ^0_{-1}2\pi\cdot x^2-\int ^0_{-1}2\pi\cdot x^3 \\ V=-2\pi\lbrack\frac{x^3}{3}+\frac{x^4}{4}\rbrack^0_{-1} \\ V=-\frac{2\pi}{12} \end{gathered}$