Question

Write down definite integrals to express the given geometric quantities :

The volume of the solid obtained by revolving $f\left(x\right)$ on $\left[a,b\right]$ around the x-axis, by the shell method.

Short answer

The volume of solid with nested shell along x-axis$=2\mathrm{\pi }{\int }_a^b\mathrm{f}\left(\mathrm{x}\right)·\mathrm{h}\left(\mathrm{x}\right)d\mathrm{x}$

Step-by-step solution

Given Information :

Given that to assume that $f\left(x\right)$is continuous and differentiable, with a continuous derivative.

Definite integral formula by shell method :

The volume of the solid S formed by revolving the region bounded by the curve $y=f\left(x\right)$between $x=a$and $x=b$about the x−axis is given by,

$V=2\mathrm{\pi }{\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right).\mathrm{h}\left(\mathrm{x}\right)d\mathrm{x}$

Where, $f\left(x\right)$and $h\left(x\right)$ are the radius and height functions for the shells in terms of x.