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 disk method.

Short answer

Volume of solid with disk cross section along x-axis is $=\mathrm{\pi }{\int }_{\mathrm{a}}^{\mathrm{b}}{\left[\mathrm{f}\left(\mathrm{x}\right)\right]}^2\mathrm{dx}$.

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 disk 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 ,

localid="1654000659093" $V=\mathrm{\pi }{\int }_{\mathrm{a}}^{\mathrm{b}}{\left[\mathrm{f}\left(\mathrm{x}\right)\right]}^{2}d\mathrm{x}$

where, cross section perpendicular to the axis of revolution is in the form of a disk of radius, $f\left(x\right)$.