Q. 1 - Calculus

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.

Step-by-Step Solution

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 1: Given Information :

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

Step 2: 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 ,

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

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